uxo burial prediction fidelity

I N S T I T U T E F O R D E F E N S E A N A L Y S E S

UXO Burial Prediction Fidelity

Jeremy A. TeichmanJenya Macheret

Shelley M. Cazares

July 2017Approved for public release;

distribution is unlimited.

IDA Document NS D-8616Log: H 17-000412

INSTITUTE FOR DEFENSE ANALYSES 4850 Mark Center Drive

Alexandria, Virginia 22311-1882

About This PublicationThis work was conducted by the Institute for Defense Analyses (IDA) under contract HQ0034-14-D-0001, Project AM-2-1528, “Assessment of Traditional and Emerging Approaches to the Detection and Classification of Surface and Buried Unexploded Ordnance (UXO),” for the Director, Environmental Security Technology Certification Program (ESTCP) and Strategic Environmental Research and Development Program (SERDP), under the Deputy Under Secretary of Defense, Installations and Environment. The views, opinions, and findings should not be construed as representing the official position of either the Department of Defense or the sponsoring organization.

For More InformationShelley M. Cazares, Project [email protected], 703-845-6792

Leonard J. Buckley, Director, Science and Technology [email protected], 703-578-2800

Copyright Notice© 2017 Institute for Defense Analyses4850 Mark Center Drive, Alexandria, Virginia 22311-1882 • (703) 845-2000.

This material may be reproduced by or for the U.S. Government pursuant to the copyright license under the clause at DFARS 252.227-7013 (a)(16) [Jun 2013].

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Executive Summary

Background IDA recently performed a numerical analysis for the Strategic Environmental Research and Development Program (SERDP) to

explore the fidelity of computational models for predicting the initial penetration depth of unexploded ordnance (UXO) in underwater sites. This briefing describes the details of our analysis.

SERDP has funded the development of many computational models to predict how underwater UXO migrates and becomes exposed over time. A munition’s initial penetration depth into the sediment is an input into these models. Other computational models have already been developed to predict the initial penetration depth of underwater mines. SERDP would like to know if and how these existing mine models could be repurposed for UXO. This issue was the focus on our analysis.

Recommendations 1. Improve sandy sediment-penetration models.

2. Develop mobility models for silt.

3. Develop mobility models for partly buried objects.

4. Develop scour burial models for silt.

5. Improve models of consolidation and creep.

6. Exercise caution in improving hydrodynamic models to support initial sediment-penetration estimates—the effect of betterhydrodynamics may be dwarfed by stochasticity due to unknown precise initial conditions at the waterline.

7. Existing sediment-penetration models (other than STRIKE35) are designed for near-cylindrical mines—for munitions,however, projectile-specific drag, lift, and moment coefficients are needed for estimating hydrodynamic stability and grossvelocity.

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8. Modules of existing depth penetration models are nearly independent and could and should be mixed and matched with littleeffort to choose best-of-breed for each phase (aero/hydrodynamic/sediment).

9. Use simplified models within the sensitivity analytical framework to understand when and how initial sediment-penetrationpredictions can be improved.

10. Use burial regime map to evaluate whether proposed improvements in model fidelity will have operational utility.

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Contents

Background ..................................................................................................................................................................................................4 Overview ......................................................................................................................................................................................................6 Useful Fidelity of Initial Burial Depth .........................................................................................................................................................8

A Notional Map .................................................................................................................................................................................15 Mobility .............................................................................................................................................................................................21 Scour ..................................................................................................................................................................................................27 Erosion ...............................................................................................................................................................................................33 Combining Effects of Mobility, Scour, and Erosion .........................................................................................................................39 Map Assumptions ..............................................................................................................................................................................43 Implications .......................................................................................................................................................................................45 Risk Buckets ......................................................................................................................................................................................47 A Quantitative Example of the Burial Regime Map .........................................................................................................................49

Achievable Fidelity of Initial Burial Depth ...............................................................................................................................................52 Hydrodynamic Phase .........................................................................................................................................................................91 Sediment Phase ..................................................................................................................................................................................99 Combine Hydrodynamic and Sediment Phases ...............................................................................................................................103

Wherre Would Additional Fidelity Help? ................................................................................................................................................112 Findings and Recommendations ..............................................................................................................................................................124 Backups ....................................................................................................................................................................................................129 References ............................................................................................................................................................................................... A-1

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Jeremy Teichman, Jenya Macheret, Shelley Cazares

Institute for Defense AnalysesScience & Technology Division

6 July 2017

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The Institute for Defense Analyses (IDA) is a not-for-profit company that operates three Federally Funded Research and Development Centers (FFRDCs). We perform scientific and technical analyses for the U.S. Government on issues related to national security.

Recently, we performed a numerical analysis for the Strategic Environmental Research and Development Program (SERDP) to explore the fidelity of computational models for predicting the initial penetration depth of unexploded ordnance (UXO) in underwater sites. This briefing describes the details of our analysis. A separate briefing provides a shorter summary. Please contact Shelley Cazares, [email protected], 703 845 6792, for the summary briefing.

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Background

Models are being built to understand how underwaterunexploded ordnance (UXO) migrates and becomesexposed over time in response to water and sedimentmotion.

Such models need initial sediment penetration estimates asinputs.

Other models have been built to estimate these initialconditions for mines dropped into water.

Can these mine models be useful for underwater UXOremediation?

What else, if anything, needs to be modified, built, ormeasured to estimate initial penetration of a munition intothe seabed floor?

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SERDP has funded the development of many computational models to predict how underwater UXO migrates and becomes exposed over time. A munition’s initial penetration depth into the sediment is an input into these models.

Other computational models have already been developed to predict the initial penetration depth of underwater mines.

SERDP would like to know if and how these existing mine models could be repurposed for UXO. This issue was the focus on our analysis.

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Overview

What fidelity (accuracy and precision) in initial sedimentpenetration is useful for underwater UXO remediation? Must feed models of mobility and exposure Erosion/accretion, scour, currents, etc.

What fidelity is achievable? Sediment penetration model fidelity Sediment property uncertainty Initial conditions for sediment impact Aerodynamic/ballistic model Hydrodynamic model

Where would additional fidelity help?

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IDA set out to answer the following questions regarding estimates of UXO initial penetration depth in an underwater environment:

What fidelity (i.e., accuracy and precision) is useful for underwater UXO remediation projects when estimating a munition’s initial penetration depth into the sediment?

What fidelity is already achievable via existing penetration models that have already been developed for underwater mines?

Where (and when) would additional fidelity be helpful for underwater UXO remediation projects?

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USEFUL FIDELITY OF INITIAL BURIAL DEPTH

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We begin with our first question: What fidelity (i.e., accuracy and precision) is useful for underwater UXO remediation projects when estimating a munition’s initial penetration depth into the sediment?

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Useful Fidelity

Important questions: Will a munition be exposed at a given time? Will a munition be mobile during a given time interval?

Important factors: Initial penetration depth (e.g., 1–50 cm)

Bottom water velocity (scour burial and mobility) (e.g., 0.5–1 m/s)

Erosion/accretion (burial/unburial) (e.g., ±20 cm)

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When determining if an underwater site must be remediated of UXO, some of the questions that the site manager may ask are:

Will a munition be exposed at any given time?

Will a munition become mobile during a given time interval?

To answer these questions, one must consider several different quantities, such as:

the munition’s initial penetration depth into the sediment,

the velocity of water at the water-sediment boundary (i.e., the bottom water velocity), and

the erosion and accretion of the sediment over time.

This analysis primarily focuses on the model’s ability to predict the munition’s initial penetration depth into the sediment. To fully consider this quantity, however, we must also explore other quantities, such as the bottom water velocity and the sediment erosion/accretion.

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Initial Stage: munition in sediment at depth

No

Yes Bottom water velocity “high”? No

YesScour to equilibrium burial depth, Immobile

Mobile, “high” risk

/2 ?Sediment floor erosion:

/2? Immobile

Immobile, “low” risk

Munition Burial Depth Analysis

Initial sediment impact

Mobility condition: /

Variation (e.g., seasonal erosion/accretion) in sediment floor vs. uncertainty in ?

Understanding the fate of a given munition

Mudline

No

Yes

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The fate of the munition after its initial penetration into the sediment is determined by the process shown in this figure. Roughly speaking:

If the initial penetration depth is deeper than the munition’s half diameter, then the munition is immobile (Rennie and Brandt 2015) and, therefore, it is stable and presents low risk.

Otherwise, the munition’s mobility will depend on the magnitude of the bottom water velocity:

– If the bottom water velocity is sufficiently high, then the munition is mobile and deemed to be high risk.

– If, on the other hand, the bottom water velocity is low, then the munition remains stationary, and the water flow around it leads to the scour process that gradually brings the munition down to the scour equilibrium depth.

o The buried munition will remain at this equilibrium scour depth if variations in sediment floor depth (i.e., erosion) take place on a much longer timescale than those of the scour process.

o If, on the other hand, the variations in sediment floor depth (i.e., erosion) take place quickly, on the timescale equal to or shorter than that of scour (e.g., due to sudden events), then the munition may become exposed and potentially mobile, with its mobility determined by the magnitude of the bottom water velocity, as discussed previously.

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Initial burial regime map: Notional sketch

Bottom water velocity,

Map is specific to: Location Depth Sediment properties Sea-bed evolution

Munition type Time period

Faster currents

Deeper initial penetration

Initi

al p

enet

ratio

n de

pth,

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A Notional Map A two-dimensional map helps us consider what fidelity in initial sediment-penetration estimates would be useful for any given

underwater UXO remediation project. In the next several slides, we discuss how we could construct this map for any given site, taking into consideration mobility, scour, and erosion:

Initial sediment-penetration conditions (position, orientation, sediment disturbance, etc.) are determined by many factors:

Local conditions (such as water depth, current, and sediment characteristics),

Conditions of impact (such as trajectory, speed, and angle of attack), and

Munition characteristics (including drag coefficient, density, shape, and size).

Currents, sediment conditions and evolution, and munition characteristics then determine the later fate of the munition.

To illuminate the relationship of initial sediment-penetration conditions to the later fate of the munition, we categorize each case by the initial penetration depth on the vertical axis and the bottom water velocity (i.e., the velocity of the water current near the water-sediment interface) on the horizontal axis. The principal fates we want to explore are exposure and mobility, both of which are driven by water currents. On this parametric landscape we then build up a series of distinguishable regions of interest. These regions of interest will help us explore what fidelity (i.e., accuracy and precision) in initial penetration models will be useful for underwater UXO remediation projects.

Note that upward along the vertical axis corresponds to deeper initial penetration into the sediment. Similarly, rightward along the horizontal axis corresponds to faster currents.

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UXO size

Faster currents

Initially exposed

Initially buried


Initi

al p

enet

ratio

n de

pth,

17

A Notional Map D is the munition dimension in the vertical direction. For typical broadside impact, this is the munition diameter. Munitions buried

deeper than D are fully buried ( . Values of 0 correspond to partly exposed munitions.

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UXO size

Recommendation #1:• Existing sediment penetration

models are focused on siltysediment.

• Sandy sediment penetrationmodels need further work.

Faster currents

Initially exposed

Initially buried


Initi

al p

enet

ratio

n de

pth,

19

A Notional Map Note that existing penetration models (developed for mines) are focused on sediments such as loose clay, silt, or mud. Because of

the greater bearing strength of sandy sediments, sinking mines typically do not self-bury in sand (Rennie and Brandt 2015), and so mine-in-sand penetration models were not needed for underwater mine remediation projects.

However, when fired, munition projectiles travel through the water with velocities sometimes much higher than terminal descent velocity. Therefore, sand-penetration models would be useful for underwater UXO remediation projects. Further development is needed to accurately predict penetration into sandy sediments.

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UXO size

Faster currents

/2


Initi

al p

enet

ratio

n de

pth,

21

Mobility The water current can dislodge munitions and carry them away from their initial impact location. Munitions buried to more than

half their diameter are generally considered to be locked-down and immobile (Rennie and Brandt 2015). Existing mobility models characterize a mobility threshold for a cylindrical object sitting upon the sediment surface (i.e., proud) in terms of parameters such as the Shields number and the surface roughness of the sediment bed (Rennie and Brandt 2015).

IDA developed its own model for predicting mobility thresholds, consistent with the phenomenological model quoted in section 4.2 of Rennie and Brandt 2015 for fully unburied munitions but extending to partly buried munitions via physics first principles. See backup slide “Mobility Model: Balance of Moments about Contact Point” for more details.

A munition’s mobility threshold is particularly important because it affects the location of the munition. In addition, once a partly buried munition is mobilized, all details of its initial impact become irrelevant, and its future fate and computation of associated risk become insensitive to its initial penetration depth.

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UXO size

Initially exposed,Mobile

Initially exposed,Immobile

Faster currents

/2

Initially exposed, Locked in place

Initially buried


Initi

al p

enet

ratio

n de

pth,

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Mobility Thus, once it is known that a munition is mobile, additional accuracy in estimating its initial penetration depth does not add value

to an underwater site manager. That is, if a munition lies anywhere under the mobility threshold curve (i.e., in the red “Initially exposed, Mobile” region of this map), it is not important to know exactly where in this region it falls. No further fidelity is needed for initial penetration models.

It is important to note the following: a time frame has now been implicitly added to this map. Water currents change with time. The water current velocity (i.e., bottom water velocity) associated with a munition on the map (i.e., its location on the horizontal axis) is the water current velocity over the time frame for initial munition mobility before anything occurs (i.e., erosion due to a sudden event) to change the state of the munition’s burial.

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UXO size

Recommendation #2:• Mobility models are developed for

sandy sediments.• Further development for mud‐like

sediments would improveknowledge of this important riskthreshold.

Recommendation #3:• Mobility models for sandy

sediments are for objects on thesea floor (proud).

• Models for partly buried objectsare needed.

Faster currents

/2Initially exposed,

MobileInitially exposed,

Immobile

Initially exposed, Locked in place

Initially buried


Initi

al p

enet

ratio

n de

pth,

25

Mobility Note that the mobility model developed by IDA (backup slide “Mobility Model: Balance of Moments about Contact Point”)

assumes there is no suction effect holding the munition to the sediment bed (e.g., our model was developed with sandy sediments in mind). However, for consolidated mud-like sediments, additional model development is needed.

Furthermore, most mobility models for sandy sediments assume the munition is proud. IDA’s model extends towards partly developed munitions. However, IDA’s model is based on first principles of physics and is therefore relatively simplistic. More sophisticated models and/or associated validation would be useful for partly buried munitions.

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UXO size

Faster currents

/2


Initi

al p

enet

ratio

n de

pth,

27

Scour Munitions less than fully buried in the sediment protrude into the bottom water currents and perturb them. The perturbed flow can

preferentially erode sediment adjacent to the munition and thereby excavate a hole into which the munition falls, further burying it. This phenomenon is known as scour burial. The scour burial process can repeat until the remaining exposed portion of the munition no longer perturbs the flow adequately to bring about further burial. We refer to this level of burial as equilibrium scour burial.

The level of equilibrium scour burial is determined by the Shields number, which represents a ratio of the force exerted on sediment particles by dynamic pressure from the circulating current to the gravitational settling force on the same particles mitigated by buoyancy. In other words, it is a ratio of perturbative force to restorative force on the sediment. At a high enough Shields number, the sediment particles will be successfully lofted into the current and removed from around the munition (Rennie and Brandt 2015). This effect occurs over the course of hours to days. If the munition is not mobile in the time frame of scour burial, then a munition less buried than the equilibrium scour burial depth will, over that timescale, reach its equilibrium scour depth, as illustrated by the upward arrows in this map.

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Initially buried



UXO size

Faster currents

/2

Initially exposed, Immobile

Buried by scour


Initi

al p

enet

ratio

n de

pth,

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Scour Due to scour, the details of the initial impact become irrelevant if the munition penetrates less than the equilibrium scour burial

depth. In other words, if we know that the munition lies anywhere under the equilibrium scour burial curve but above the mobility threshold curve, in the “Initially exposed, Immobile, Buried by scour” region of the map, then we do not need to know exactly where in this region it lies. That is, no further fidelity in initial penetration models is needed.

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UXO size

Recommendation #4:• Scour burial models are developed

for sandy sediments.• Scour burial models in silty

sediments, if applicable, needdevelopment.

Faster currents


Initially buried

/2


Buried by scour


Initi

al p

enet

ratio

n de

pth,

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Scour Note that scour burial models are built for sandy sediments. For muds and clays, such models need development, if relevant.

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UXO size

Maximum erosion

Faster currents


/2

ErosionMudline


Buried by scour

Initially buried,Exposed by erosion

Initially buried,Not exposed by erosion


Initi

al p

enet

ratio

n de

pth,

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Erosion Over days to decades, the sediment in the vicinity of the munition may experience erosion or accretion as material is removed from

or deposited into the neighborhood. The seabed may also experience gross motion, such as migration of formations such as sand bars or mega-ripples. The effect of such erosion/accretion is to alter the degree of burial of the munitions. Other than in energetic events such as storms causing sudden large changes in the sediment, the pace of erosion and accretion is slower than the timescale for scour burial. Therefore, one would expect that such munitions would never be other than transiently more exposed than their equilibrium scour burial depth.

We denote the maximum erosion by , noting that, again, the specification of implicitly captures a specific time duration, and so our map implicitly characterizes a particular location over a particular period of time. Munitions initially buried deeper than this maximum erosion will not, over the time duration of interest, be at all exposed. Other munitions will be durably exposed to a degree limited by scour burial.

Our investigation of the literature suggests that the mobility threshold will fall completely below the equilibrium scour burial curve (computed based on mobility thresholds and scour burial models presented in Rennie and Brandt 2015), as depicted in our notional map. Consequently, other-than-energetic erosion events are unlikely to mobilize previously immobile munitions. Rather, scour burial will always keep pace with gradual erosion maintaining the immobility of the munitions. That is, considering the following expressions from Rennie and Brandt 2015:

Mobility threshold: 1.2 . . 1 , where U is the bottom water velocity, g is gravitational acceleration, D is the munition diameter, Smunition is the munition specific gravity, and d is the bottom roughness (grain diameter).

Conditions for 100% equilibrium scour burial: .

, where Ssediment is the sediment particle specific

gravity, and f is a friction factor.

We then evaluate the ratio of the two bottom water velocity thresholds: 3.5.

For munition specific gravities at least as high as sediment specific gravities, and for sediment particle diameters at most 1/60 the munitions diameter (e.g., 10 cm munition, 1 mm sand), the mobility threshold will exceed the threshold for 100% scour burial for friction

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factors greater than 0.05. Smaller sediment particles and denser munitions each further increase the mobility threshold relative to the 100% scour burial threshold (i.e., mobility requires a higher velocity than full scour burial at friction factors below 0.05).

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UXO size

Maximum erosion Recommendation #5:• Munitions in silty sediments sink over

time (months to decades) due to:• Consolidation (excess pore pressure

relief)• Creep (long time‐scale viscous

behavior)• Consolidation and creep models need

development, since these effectscounteract erosion exposure.

Faster currents


/2

ErosionMudline



Buried by scour

Initially buried,Exposed by erosion

Initially buried,Not exposed by erosion


Initi

al p

enet

ratio

n de

pth,

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Erosion Effects not captured in this map are the long-term behaviors of muddy or clay-like sediments:

Consolidation refers to the gradual (hours to years) relief of excess pore pressure in the sediment. The forceful entry of the munition into the sediment locally increases pressure in the sediment. Over time, fluid flow through the porous structure of the sediment relieves this excess pressure. The munition can sink further as the resistance exerted by the higher pore pressure abates.

Such sediments also exhibit viscous behaviors over long timescales, flowing very slowly to relieve applied stresses. This is known as creep. A munition denser than the surrounding sediment will slowly continue to sink into the sediment over years to decades until it reaches density equilibrium.

Any complete picture of munition evolution in sediment would need to account for these effects. More relevant to the present topic: these effects (and uncertainty in these effects) will diminish the sensitivity of the position evolution of the munition to its initial penetration conditions. These effects will also diminish any leverage to improve estimation of such evolution via improving prediction of the munition’s initial penetration depth.

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Exposed by erosion

Exposed

Mobilized by erosion

Always buried



Initi

al p

enet

ratio

n de

pth,

UXO size

mobility threshold

Maximum erosion

MobileFaster currents

/2


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Combining Effects of Mobility, Scour, and Erosion Combining the effects discussed so far, we can partition the map into a series of regions relating to distinguishable categories of

munition fate as a function of its initial sediment-penetration depth and the bottom water velocity. In most cases, knowing in which region a munition lies is important, but exactly where in that region is irrelevant:

At the top in green (“Always buried”) are munitions buried deeper than the reach of erosion in the time frame of consideration. (Remember, the higher up the vertical axis, the deeper the initial penetration into the sediment.) These munitions will remain continuously buried for the duration of the interval.

On the bottom right in red (“Mobile”), munitions are moved over a short period of time from their initial location.

Above that region, in orange (“Mobilized by erosion”), are munitions that would be mobilized by sufficiently rapid erosion. Typically, however, erosion occurs more slowly than scour burial. Therefore, such munitions, if initially exposed, will be scour-buried shortly after initially reaching the sea floor. Otherwise, if not initially exposed, then scour burial will keep them below the sediment surface in the face of gradual erosion.

On the lower left in yellow (“Exposed”) are munitions partly buried upon impact in waters sufficiently quiescent to leave the munition exposed even after scour effects (i.e., even after reaching the scour equilibrium burial depth). Accretion may fully bury such munitions, but subsequent erosion will leave them exposed. If, over time, erosion relative to the initial sediment level is on the order of the munition size, then irrespective of where in this region a munition initially impacted the sediment, it will end up at its equilibrium scour burial depth during the period of maximum erosion, thus effacing all details of the initial impact within this region. In other words, there is a ratcheting effect—as erosion diminishes the instantaneous burial depth below the equilibrium scour burial depth, scour will restore the equilibrium scour burial depth. Subsequent accretion may bury the munition further, but successive erosion/accretion cycles can only bury the munition deeper.

The final region, shown in blue (“Exposed by erosion”), second from the bottom on the left and extending to the center bottom, encompasses munitions that were either initially buried or scour buried soon thereafter, but, in either case, not so deeply buried as to be beyond the reach of erosion.

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Tran

sien

t exp

osur

e by

er

osio

n

Durable exposure by erosion

Exposed

Mobilized by erosion

Always buried



UXO size

mobility threshold

Maximum erosion


/2


Initi

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ratio

n de

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Combining Effects of Mobility, Scour, and Erosion It is worthwhile to further subdivide this last region:

The blue (“Durable exposure by erosion”) region second from the bottom on the left captures munitions initially fully buried but then exposed by erosion. Because the bottom currents in this region of the map are insufficient to fully scour bury the munitions subsequent to exposure by erosion, such munitions will remain durably exposed. Further, such munitions will not remain more exposed than their equilibrium scour burial depth.

The narrow vertical region in the center of the parameter space shown in purple (“Transient exposure by erosion”) represents munitions that become fully buried over the scour burial timescale even if they were initially exposed. Any erosion to the point of exposure will be mitigated by full scour burial, so although such munitions may surface periodically on the sediment bed, such exposure will be transient subject to scour burial. Nevertheless, barring subsequent accretion, these munitions will remain very shallowly buried, so they may be amenable to detection and remediation efforts.

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Initial burial regime map: Our assumptions

Constant bottom water velocity But in reality, varies with time, and sudden erosion events (e.g.,

storms) are highly correlated with increases in .

Mobility time < scour burial time < erosion time But for some events (e.g., storms): erosion time < scour burial time

Independent processes But storm events may be

correlated with seasonalerosion/accretion cycle

Deterministic evolution But in reality, is stochastic

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Map Assumptions In constructing our parametric partition of this map, we made a series of simplifications. Here we elaborate on when those

simplifications might be invalid:

First, we treated bottom water velocity as time-invariant. Of course, in reality, bottom water velocity varies with time, weather conditions, tides, and more.

Also, typically, erosion occurs more slowly than scour burial, which, in turn, occurs more slowly than movement of munitions by the current. However, storm events, which may be the greatest source of exposure and mobility risk, violate these assumptions. During storm events, bottom water velocity may increase dramatically, whether due to enhanced currents or due to wave-induced circulation at the seabed. The same effects can create rapid erosion at a timescale shorter than that for scour burial. Thus, during storm events, munitions either insufficiently buried or suddenly exposed by erosion may be mobilized by the temporarily enhanced bottom current.

Bottom water velocities may also vary seasonally in synchrony with periodic erosion and accretion cycles as well as storm events. Our analysis does not capture the correlation of these processes.

Bottom water velocity and erosion/accretion are stochastic variables whose randomness is not captured by our deterministic analysis. Our map shows a single maximum degree of erosion . The stochastic nature of adds uncertainty to predictions of munition fate, and assessments of risk probabilities would need to account for the statistical distribution of

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Initial burial regime map: Implications

Burial regime maps are scenario specific. Localization to a sector of the map indicates what

level of initial burial fidelity is useful: High fidelity may be

useful near edges ofsectors

High fidelity may not beas useful within a sector

Useful fidelity variesnot only with localconditions, but alsowith burial depth

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Implications Every scenario would require its own regime map to capture the local sediment conditions, water depth, currents, erosion profile,

and munitions of interest. For each such scenario, the regime map would facilitate considerations of required prediction fidelity for initial sediment penetration. As discussed, identifying in which region of this map the munition lies would be useful, and therefore, high-fidelity initial predictions of penetration depth near the region boundaries may be very useful in some cases. However, far from region boundaries, high-fidelity prediction of initial penetration depth may have little influence on the risk associated with the munition.

The critical observation is that the there is no universally useful fidelity for penetration prediction. The useful fidelity varies with penetration depth itself as well as with the local conditions, which determine the locations of the regime boundaries.

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Initial burial regime map: Risk buckets

Exposed

Transient exposure

Always buried


UXO size

mobility threshold

Maximum erosion


/2

Initi

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enet

ratio

n de

pth,


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Risk Buckets One could further simplify the regime map by dividing it into what we call risk buckets. Some of the distinguishable fates on the

full regime map do not translate into different risks. It may principally be useful to distinguish levels or categories or “buckets” of risk rather than simply distinct fates. This diagram of risk buckets combines the regimes of initially exposed munitions and those durably exposed by erosion into a single risk bucket of stationary munitions exposed for sustained durations. This diagram also combines the map regimes corresponding to transient exposure by erosion and mobilization by erosion. Since those munitions above the mobility curve would typically be reburied by scour before they could be mobilized, in the absence of events generating rapid erosion and high currents, both regimes correspond to munitions temporarily exposed by erosion and remaining shallowly buried thereafter. In that vein, one could equally well think of the four risk buckets as:

exposed,

deeply buried,

shallowly buried, and

mobile.

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Initial burial regime map: Sandy Sediment Example

Broad side impact:Water depth: uniform distribution, 3 to 5 mWater impact velocity: uniform distribution, 50 to 300 m/sWater impact angle: normal distribution, 45 15°Sediment shear strength: normal distribution, 20 5 kPa

Fraction of UXO

Potentially mobile UXO

0 2 4 6 8 100

0.1

0.2

0.3

0.4mobility lineequilibrium scour depthUXO diameter95% of UXO burial depth5% of UXO burial depth

buttom current velocity, m/s

initi

al b

uria

l dep

th, m

Uncertainty in penetration depth due to variation in water impact angle and velocity, water depth, and sediment shear strength

Initial pen

etratio

n depth,

(m)

Bottom water velocity, (m/s)

%

%

mobility thresholdequilibrium scour burial

D

/2

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A Quantitative Example of the Burial Regime Map Here, on the right, we show a quantitative instance of the mobility and scour burial regimes of the map for a particular scenario.

We utilized the equilibrium scour burial depth model from Rennie and Brandt 2015 and the IDA mobility threshold model for 100 micron sand (backup slide “Mobility Model: Balance of Moments about Contact Point”). On the left, we show a histogram of initial sediment-penetration depths from a Monte Carlo model discussed later in this briefing. The Monte Carlo simulation randomized over input values as described at the top of this slide. As shown, although the mean initial penetration depth is approximately half the munition diameter, the variability in the penetration depth in this case is large compared with the scale of the regimes on the map. In this case, the benefit of predicting initial penetration depth to high fidelity for any given input conditions would be negated by the high variability in initial penetration depth due to input condition variation. In short, the useful fidelity of initial penetration prediction is bounded by the variability in uncontrolled and unknown initial conditions.

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Recommendations: Sediment types

Sandy Loose sediment with high bearing

strength Scour burial: Local enhancement

of eddy currents further burymunitions

Recommendation #1: Improvesandy sediment penetrationmodels.

Recommendation #3: Developmobility models for partly buriedobjects.

Cohesive (clay, mud) Cohesive sediment with low bearing

strength Consolidation (slow relief of pore

pressure): Munitions sink over hours toyears as excess water pore pressureequilibrates

Creep (viscous behavior at long timescales): Munitions continue to sinkover years to decades

Recommendation #2: Developmobility models for cohesive sediment.

Recommendation #4: Develop scourburial models for cohesive sediment.

Recommendation #5: Improveconsolidation and creep models.

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Here we consolidate the recommendations found in the preceding section.

52

ACHIEVABLE FIDELITY OF INITIAL BURIAL DEPTH

53

We have now explored what fidelity is useful for underwater UXO remediation projects when estimating a munition’s initial penetration depth into the sediment.

Next, we will consider our second question: What fidelity is already achievable via existing penetration models that have already been developed for underwater mines?

54

Predicting Penetration of Munitions into the Seabed

Aero Phase

Hydro Phase

Sediment Phase

• Water Impactvelocity vector

• Angle of attack

• Water depth

• Munition drag coefficient

• Sediment shearstrength, density,bearing strength

• Rate dependenceof resistance force

• Sediment Impactvelocity vector

• Angle of attack

Initial Burial depth

MINE6D(MIT)

STRIKE35(NPS)

Existing Models

IMPACT35(NPS)

IMPACT28(NRL)

Modeling Objective:Estimate initial sediment penetration of UXO for inputs to models of burial, exposure, and mobility over time with the ultimate goal of predicting risk of UXO exposure and location.

Key parameters

Figure adapted from P.C. Chu et al., “Mine Burial Prediction Experiment,” J. of Counter‐Ordnance Technology (5th Int. Symp. On Technology and Mine Problem).

55

The descent of a munition to its final resting position in the sediment proceeds in three phases:

AERO PHASE—For a munition released above the water, there is an aerodynamic or ballistic phase in which the munition, under the influence of gravity and aerodynamic forces and moments, travels from its release point to its water entry point. The initial conditions for the aerodynamic phase are given by the delivery mechanism of the weapon. For a mortar, it might be a muzzle velocity and elevation angle. For a bomb, it might be a release altitude and aircraft velocity. The aerodynamic model follows the projectile through the air and gives its location, orientation, velocity, and dynamics when it hits the waterline. For projectiles designed for aerodynamic stability, the orientation is fixed relative to the trajectory angle with small perturbations. These perturbations, however, can be amplified by asymmetric forces as the projectile transitions into the water.

HYDRO PHASE—Once the projectile begins entering the water, there are interface effects at the waterline. Especially at high velocities, air becomes entrained along with the projectile and enters the water as a cavitation bubble surrounding the projectile. Drag forces, moments, and buoyancy are all affected by the cavitation bubble. The projectile travels through the water column influenced by gravity, buoyancy, and hydrodynamic forces and moments produced by viscous interaction with the water, as well as inertia of the water disturbed by the transit. If an aerodynamically stable projectile enters the water without damage (e.g., with intact fins) and with sufficiently low angle of attack, it may remain hydrodynamically stable during its water transit. A hydrodynamic model follows the projectile from its initial conditions given by the output of the aerodynamic model through its transit of the water column and provides the position, orientation, velocity, and dynamics of the projectile when it hits the so-called mudline, the interface between the water and the sediment.

SEDIMENT PHASE—The sediment model then takes as its initial conditions the output of the hydrodynamic model and tracks the projectile’s penetration into the sediment, accounting for gravity and phenomena-generating forces resisting sediment penetration, potentially including buoyancy, friction, inertia, sediment-bearing strength and shear strength, viscosity, pore pressure, etc. These properties may vary with location and depth below the mudline and may themselves be rate dependent. The sediment model tracks the projectile until it comes to a halt and outputs the final resting orientation and position of the projectile, most notably, its penetration depth below the mudline.

In addition to properties of the media, all these models require parameters describing the projectile such as its density and drag, lift, and moment coefficients.

56

Aerodynamic models generally exist within the ballistics community. These models are necessary because most projectiles do not achieve aerodynamic terminal velocity, so the water-entry velocity must be computed dynamically. Also, the trajectory is not generally perfectly vertical.

One of the primary thrusts of the analysis described in this document was the applicability or extensibility of models from the underwater mine community to the underwater UXO problem. The four models we considered are shown on the lower right of the slide. The ensuing slides will discuss these models further, but briefly: MINE6D developed by the Massachusetts Institute of Technology (MIT) and STRIKE35 developed by the Naval Postgraduate School (NPS) cover only the hydrodynamic phase, and IMPACT28 from the Naval Research Laboratory (NRL) and IMPACT35 from NPS cover the hydrodynamic and sediment-penetration phases as well as a rudimentary treatment of the aerodynamic phase.

58

4P. Chu, et al., “Modeling of Underwater Bomb Trajectory for Mine Clearance,” Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 8 (1) (2011): 25–36.

3P. Chu and C. Fan, “Mine‐Impact Burial Model (IMPACT‐35) Verification and Improvement Using Sediment Bearing Factor Method,” IEEE J. of Ocean Eng.. 32, no. 1 (January 2007 ).

1P. Chu, “Mine Impact Burial Prediction From One to Three Dimensions,” Applied Mechanics Review62 (January 2009).2J. Mann et al., “Deterministic and Stochastic Predictions of Motion Dynamics of Cylindrical Mines Falling Through Water,” IEEE Journal of Ocean Engineering32, no. 1 (2007).

Brief Description of the ModelsModel Name Description Validation

2D models predict rigid body motion in the x–z plane including rotation about the y-axis.

IMPACT25/28 3 DOF model from air-drop through water column.1• Hydro: Developed to address rotation (absent in 1D predecessor IBPM).• Sediment: Handles multi-layers.

Limited low velocity mine drops (<10 m/s)

3D models predict rigid body motion in x,y,z coordinates including rotation about the x,y,z axes. All have been compared favorably to limited experimental results/

MINE6-D(Hydrodynamic only)

6-DOF model2 designed to accurately capture complex 3D dynamics of minesin the water column.


IMPACT35 5-DOF model3 (neglects rotation about the axis of symmetry) for nearcylindrical mines.• Hydro: Designed to account for y-direction currents and observed 3D

dynamic modes of falling mines.• Sediment: Adds new rate-dependent sediment strength model to IMPACT

25/28.


STRIKE35(Hydrodynamic only)

6-DOF model describes motion of JDAM through a water column.4Designed to model high velocity hydrodynamic behavior of a tapered objectwith fins.

Several high velocity JDAM drops (400 m/s)

Models and experiments show high sensitivity to initial conditions at waterline.

59

This slide provides a compare-and-contrast of the four models introduced on the previous slide:

IBPM, the Interim Burial Prediction Model, was a one-dimensional (1D) model that assumed a constant mine orientation. IBPM was expanded into a two-dimensional (2D) model capturing three degrees of freedom (two directions in position and one orientation, all in a single 2D plane). That model was implemented in BASIC as IMPACT25 and in MATLAB as IMPACT28. The IMPACT25/28 model addresses all three phases of descent (aerodynamic, hydrodynamic, and sediment) and accommodates multiple sediment layers with different properties. IMPACT25/28 utilizes a simplified sediment-penetration model based on bearing strength and assuming a simple relationship between bearing strength and shear strength. IMPACT25/28 also incorporates inertial resistance of the sediment, which may not be of critical importance in low-velocity mine sediment impacts but could be essential in high-velocity munition projectile impacts. IMPACT25/28 includes a cavitation model capturing such aspects as cavity oscillation frequency. Although capturing a lot of cavitation detail, the model requires empirical determination of a number of coefficients.

IMPACT35 was then developed to incorporate more complex three-dimensional (3D) observed hydrodynamic behavior in descending mines, such as helical motion. A third dimension is also required to account for water currents in directions other than in-plane with the falling projectile. IMPACT35 is a five-degree-of-freedom model incorporating three directions of translation and two of rotation (pitch and yaw, but not roll, which is assumed to be immaterial due to rotation symmetry of the mines under consideration). IMPACT35 implemented what is known as the delta method for sediment-penetration estimation, which incorporates the effect of pore pressure. Further analysis and experiments with low-velocity mine drops suggested a better sediment-penetration model known as the bearing factor method, which was later incorporated into IMPACT35 in lieu of the delta method.

MINE6D was developed for the same reasons as IMPACT35: to accurately capture the complex 3D dynamics of sinking mines. MINE6D adds the sixth degree of freedom of roll and can accommodate rotationally asymmetric shapes. MINE6D attempts to more rigorously capture the hydrodynamic interaction of the water with the sinking mine.

All three of the preceding models were supported by limited validation experiments with low-velocity mine drops. In mine drop experiments, the mines entered the water gently or were dropped from a low elevation. They typically, therefore, entered the water below hydrodynamic terminal velocity and accelerated as they descended. Contrast this with the UXO problem in which munition projectiles enter the water at velocities far in excess of the hydrodynamic terminal velocity and decelerate as they descend.

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Hydrodynamic terminal velocity is the velocity that is asymptotically approached (from above or below) at which the hydrodynamic resistance forces exactly balance buoyancy and gravity to produce zero acceleration. Therefore:

STRIKE35 was produced to capture the behavior of the high-velocity projectiles. It was developed/validated in conjunctionwith a set of experiments dropping Joint Direct Attack Munitions (JDAMs) into an instrumented pond. The JDAMs enteredthe water at roughly 400 m/s. STRIKE35 utilizes simple forward Euler time discretization to model progress through thewater. Where STRIKE35 differs most from the other models is in its use of projectile-specific semi-empirical expressions forhydrodynamic lift, drag, and moment coefficients accommodating specifics such as fin geometry. Any of the models could beenhanced with experimentally derived hydrodynamic coefficients at the velocity ranges of interest. STRIKE35 takes as itsinitial conditions the entry velocity into the water and does not incorporate sediment penetration. STRIKE35 takes an ad hocapproach to cavitation by assuming that cavitation reduces the drag by a factor of 10 and then suddenly ceases when themunition angle of attack exceeds a pre-specified value.

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Recommendation #6:Exercise caution improving hydrodynamic models. Effect of better hydrodynamics may be

dwarfed by stochasticity due to unknown precise initial conditions at the water line.

Brief Description of the ModelsModel Name Description Validation

2D models predict rigid body motion in the x-z plane including rotation about the y-axis.

IMPACT25/28 3-DOF model from air-drop through water column.1• Hydro: Developed to address rotation (absent in 1D predecessor IBPM).• Sediment: Handles multi-layers


3D models predict rigid body motion in x,y,z coordinates including rotation about the x,y,z axes. All have been compared favorably to limited experimental results.

MINE6-D(Hydrodynamic only)

6-DOF model2 designed to accurately capture complex 3D dynamics of minesin the water column.


IMPACT35 5-DOF model3 (neglects rotation about the axis of symmetry) for nearcylindrical mines.• Hydro: Designed to account for y-direction currents and observed 3D

dynamic modes of falling mines.• Sediment: Adds new rate-dependent sediment strength model to IMPACT

25/28.


STRIKE35(Hydrodynamic only)

6-DOF model describes motion of JDAM through a water column.4Designed to model high velocity hydrodynamic behavior of a tapered objectwith fins.

Several high velocity JDAM drops (400 m/s)

Models and experiments show high sensitivity to initial conditions at waterline.

4P. Chu, et al., “Modeling of Underwater Bomb Trajectory for Mine Clearance,” Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 8 (1) (2011): 25–36.

3P. Chu and C. Fan, “Mine‐Impact Burial Model (IMPACT‐35) Verification and Improvement Using Sediment Bearing Factor Method,” IEEE J. of Ocean Eng.. 32, no. 1 (January 2007 ).

1P. Chu, “Mine Impact Burial Prediction From One to Three Dimensions,” Applied Mechanics Review 62 (January 2009).2J. Mann et al., “Deterministic and Stochastic Predictions of Motion Dynamics of Cylindrical Mines Falling Through Water,” IEEE Journal of Ocean Engineering 32, no. 1 (2007).

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These four models may do an excellent job predicting the progress of a projectile through the water given initial conditions. However, for the munition projectiles of interest in underwater UXO remediation, the initial conditions at the waterline may be highly uncertain. Therefore, adding fidelity to the hydrodynamic model may not address the dominant sources of uncertainty in the sediment-penetration prediction. In addition, the details of the projectile’s progress through the water column may not have a strong influence on its initial conditions at the mudline and may therefore not strongly influence its sediment-penetration depth. Experiments and models have exhibited strong sensitivity to initial conditions. In experiments with nominally similar mine-release conditions, different falling modalities (e.g., see-saw vs. helical spiral) have been observed, suggesting instabilities not conducive to modeling. The statistics of the mudline impact velocity are more important than identifying which precise initial conditions lead to each descent modality.

Ultimately, caution should be exercised before investing further in high-fidelity hydrodynamic models as the goal is to characterize sediment-penetration depth.

64

Model Features

Dimen

sions

DOF

Aerodyna

mic

Wind

Waterline

interface

Hydrodyna

mic

Cavitatio

n

Fins

Current

Mud

line

interface

Sediment

Inertia

l drag

IMPACT25/28 2 3 Y N Y Y Y N N Y Y Y

IMPACT35 3 5 Y Y Y Y N N Y Y Y N

STRIKE35 3 6 N Y Y Y Y N N N

MINE6D 3 6 N Y Y Y N Y N N

65

Here we show a tabular breakdown of the features captured by the various models.

66

Model Features

Dimen

sions

DOF

Aerodyna

mic

Wind

Waterline

interface

Hydrodyna

mic

Cavitatio

n

Fins

Current

Mud

line

interface

Sediment

Inertia

l drag

IMPACT25/28 2 3 Y N Y Y Y N N Y Y Y

IMPACT35 3 5 Y Y Y Y N N Y Y Y N

STRIKE35 3 6 N Y Y Y Y N N N

MINE6D 3 6 N Y Y Y N Y N N

Recommendation #7:Models (other than STRIKE35) are designed for near‐cylindrical mines.

For UXO, projectile‐specific drag, lift, and moment coefficients are needed for estimating hydrodynamic stability and gross velocity.

Recommendation #8:Model modules (aero, hydro, sediment) are nearly independent and could and should be mixed and matched with little effort to choose best‐of‐breed for each regime (aero, hydro,

sediment) for each scenario of interest.

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Even though STRIKE35 is the only model to use highly specific empirical hydrodynamic coefficients, it would not be difficult to implement this type of characteristic in any of the hydrodynamic models. The hard work is gathering the empirical data on each class of projectile. If the models other than STRIKE35 are used, the existing hydrodynamic coefficients based on cylindrical mines would likely not adequately capture the hydrodynamic behavior of munition projectiles for good sediment-penetration prediction. In particular, the models ought to capture the hydrodynamic stability of the projectiles, because projectiles penetrating the water column in a stable fashion will exhibit much higher terminal velocities and higher sediment impact velocities than do mines.

Also of note, the models incorporating multiple phases of descent (aerodynamic, hydrodynamic, and sediment) are spliced together in a relatively compartmentalized fashion, allowing the mixing and matching of modules between models with relative ease.

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*P. Chu, “Mine Impact Burial Prediction From One to Three Dimensions,” Applied Mechanics Review 62 (January 2009).

Questions: • Is this improvement operationally significant?• How much further improvement is possible?• How much further improvement would be operationally useful?

Low‐velocity (<10 m/s) mine impacts. Mines were gently deposited in the water and reached terminal velocity in the water.

Comparing 2D and 3D Burial Prediction models*

Model Description Used in

Bearing strength

Naïve weight bearing capacity relationship to shear strength; Inertial drag included

IMPACT25/28

Delta method

Includes shear forces, buoyancy, and water pore pressure

IMPACT35 early version

Bearing factor method

Found to be more accurate than Delta method in low velocity mine experiments

IMPACT35 later version

IMPACT35 matched experimental results more closely than IMPACT28.

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This slide compares a 2D model (IMPACT28) and a 3D model (IMPACT35) for estimating the penetration depth of mines.

The pertinent parameters for the 2D model (IMPACT28) are sediment density and bearing strength. For sediment resistance, the 2D assumes a bearing strength 10 times that of shear strength. The 3D model (IMPACT35) uses the Delta or Bearing factor method:

The Delta method assumes that the mine pushes the sediment and leaves space in the wake as it penetrates the sediment. This space is refilled by water and the water cavity is produced.

The Bearing factor method is based on the fact that the shear resistance, which retards the mine propagation, is proportional to the product of the sediment shear strength and the rupture area perpendicular to the velocity with a non-negative bearing factor.

(It was shown that the Bearing factor method gives more accurate predictions than the Delta method.)

The bar chart shown in this slide compares the predictions of the 2D model (IMPACT28, light gray) and 3D (IMPACT35, dark gray) with experimental data (medium gray). The 2D model (IMPACT28) overpredicts the actual sediment-penetration depth by an order of magnitude on average. As can be seen from the chart, the 3D model (IMPACT35) predictions are in better agreement with the experimental data.

It is important to note, however, that the 2D model errors are on the order of 2 to 3 cm, and the question we pose here is: How significant is the improvement in model predictions?

Realizing that there is considerable uncertainty in many other model parameters as well as initial conditions of the mine impact into sediments, we would like to understand to what extend the improvement in modeling predictions can be actually meaningful.

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Applying Existing Models to Predict Location of UXO*

NRL Stennis developed a prototype framework for risk assessment1. Aerodynamic model – NRL in-house2. Hydrodynamic model – IMPACT35 + STRIKE353. Sediment penetration model – IMPACT354. Mobility and migration models – UXO MM

They incorporated this framework into a Bayesian network

They used a notional Camp Perry scenario to show that they couldgenerate outputs based on a mash-up of pieces of the various codes ina way that could be compared to survey data Their work demonstrates a proof-of-principle of incorporating existing mine

models together for munition penetration prediction Their work has not yet been fully validated with empirical data

*K. Todd Holland, “A Wide Area Risk Assessment Framework for Underwater Military Munitions Response,” MR2411 Progress Review, May 2015.

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NRL Stennis used several models in tandem to construct a prototype probabilistic framework to statistically categorize spatial distribution of expected initial munitions contamination (Holland 2015). The following modules were assembled together:

aerodynamic trajectory

impact with the air-water interface

free-fall through the water column (Chu and Ray 2006) – IMPACT35

impact penetration with a sedimentary seabed (Chu and Fan 2007) – IMPACT35

mobility and migration models (Wilson et al. 2008) – UXO MM

The easting/northing predicted location of the munitions were shown to be in close agreement with available data, which is the density of the UXO “targets” found by an underwater magnetometer survey. The conclusion of this work is that IMPACT35 could be quickly adapted for use in underwater UXO remediation projects, with the addition of ballistic equations and range firing records to simulate impact penetration and to predict a buried UXO population distribution.

Note, however, that to date, the results are limited to validating the easting/northing location estimates of the initial munition impacts, not their penetration depth into the sediment nor their subsequent mobility.

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Model dependencies

Aerodynamic model Outputs: , ,

Hydrodynamic model Inputs: , , , Outputs: , ,

Sediment penetration model Inputs: , , , Output:

Other properties Air properties, water

properties, current, wind,munition properties (lift, drag,density, shape, size)

initial penetration depthvelocity at mudlineelevation angle vs. mudlineangle of attack at mudlinesediment shear strengthwater depthvelocity at waterlineelevation angle vs. waterlineangle of attack at waterlinemunition massmunition lengthmunition diametermunition drag coefficient

, ,

, , , , , ,General Munition Burial Model:

Waterline

Mudline

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Here we begin a mathematical characterization of the fidelity of initial sediment-penetration depth predictions. (Note that by “initial” sediment penetration, we are referring to the munition’s sediment penetration upon impact. We are not referring to the munition’s sediment penetration due to subsequent mobility, scour, or erosion/accretion.)

Our goal in this exercise is to build a mathematical framework for quantifying the relative contributions of various sources of initial sediment-penetration prediction error. This framework allows us to then explore the comparative value of reducing those error sources. We sought to identify which sources of error were dominant vs. negligible, as well as which sources of error were reducible vs. irreducible.

We start with a functional dependency model. The initial sediment-penetration depth depends on the initial conditions of the munition’s trajectory (i.e., at the water line), the environment, and the munition’s properties:

Initial conditions of munition trajectory: The moment in time to which the initial conditions are referenced is arbitrary. One could consider the functional model to begin when the projectile is released/expelled from its launching platform. However, it may instead be convenient to consider initial conditions at the moment the munition projectile first contacts the waterline from the air. We denote conditions associated with this first waterline contact by a “00” subscript. The munition’s initial conditions are thus its position, orientation, velocity, and rotation rate:

– Velocity at the waterline is described by a speed and trajectory angle .

– Angle of attack at the waterline is denoted by and captures the munition’s orientation at the waterline relative to its trajectory angle . There could also be a yaw angle, but for simplicity we omit that here along with initial rotation rates.

– Position at the waterline will only affect sediment-penetration depth via local environmental conditions, and therefore we need not explicitly consider the munition’s position at the waterline.

Environment: We can capture the local environmental conditions via several parameters. Here we use two parameters, omitting others for clarity:

– water depth and

– sediment shear strength .

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Munition properties: We can characterize the munition via several parameters, as well. Here we omit many of thoseparameters for clarity (such as the munition’s mass distribution, center of pressure, detailed shape, lift and momentcoefficients, and so forth) and include only its:

– mass ,

– geometry (length and diameter ), and

– drag coefficient .

Bearing all these parameters in mind, the functional model simply specifies the output (the initial sediment-penetration depth ) as a function of the inputs. In this case, , , , , , , , , .

The various model regimes (i.e., aerodynamic, hydrodynamic, sediment) link the initial and final conditions (inputs and outputs) in each domain:

The aerodynamic model takes as its input the munition projectile’s release conditions and outputs the “00” conditions at thewaterline.

In turn, these “00” conditions serve as the inputs to the hydrodynamic model, which then computes the “0” conditions at themudline.

The “0” conditions then feed the sediment-penetration model, which outputs the initial sediment-penetration depth .

Because no model perfectly represents the physical world, even with perfectly correct inputs, the output of every model will differfrom what an experiment would show. We refer to this as the model error:

for the hydrodynamic model error and

for the sediment model error.

We do not fully consider the aerodynamic model here; we take the “00” waterline initial conditions as the point of entry into ourconsideration.

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Sensitivities of General Munition Burial Model

Sediment penetration model:

model sensitivity Hydrodynamic model:

Combined model

error

Model error

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In addition to the model errors and , any implementation of the model will provide imperfect results because the inputs can never be perfectly well known. The sensitivity of the model is the degree to which the outputs are affected by variation (or error) in the inputs. For a high-fidelity model, the sensitivity of the model may be closely approximated by the sensitivity of the underlying physical system to perturbations in the initial conditions. Assuming that the outputs vary smoothly with the input values, for sufficiently small deviations in initial conditions, a linear model describes the sensitivities in terms of derivatives of the functional model. Adopting the nomenclature that is the error in parameter , we denote, for example, the error in the input initial trajectory angle at the waterline as . Thus, in the hydrodynamic model, for example, the contribution to the error in the prediction of mudline velocity from error in the input waterline velocity would be given by the product of the input error and the sensitivity . This framework can be

constructed for either single-domain models or for the end-to-end functional model.

The overall output prediction error is given by the sum of the modeling error and the contributions from input errors. The input errors to one domain can be constructed from the input errors to the preceding domain and the modeling error in the preceding domain.

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Errors and Assumptions

, , : from ballistic/aerodynamic uncertainty.

: Short term: from waves, tide. Long-term: from water level, mudline evolution.

: from measurement uncertainty, spatial variation, time evolution

Waterline

Mudline

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Since our consideration begins at the waterline, we consider the errors in waterline initial conditions to be overall inputs to our system. These errors come from unmeasured variation in the firing and transit conditions of the projectiles as well as errors in the aerodynamic model.

Errors in the water depth can arise from short-term variability in water depth due to waves and tides. A projectile impacting the crest of a wave would experience a different water column depth than a projectile impacting the trough of the wave. Water depth also varies over long periods of time due to changes in overall absolute water level as well as evolution of the sediment contour (i.e., erosion, accretion, formation movement).

Errors in sediment properties such as shear strength arise from difficulty making high-fidelity measurements of sediment properties as well as variability over time and from point to point within a region of interest.

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Errors and Assumptions

, , : from ballistic/aerodynamic uncertainty.

: Short term: from waves, tide. Long-term: from water level, mudline evolution.

: from measurement uncertainty, spatial variation, time evolution

→Statistical characterization of error:

error in one instanceaverage magnitude of error

Waterline

Mudline

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Some of these sources of error are reducible and some are irreducible. For example, sediment property measurement error might be reduced by development of improved measurement techniques and equipment. On the other hand, for retrospective studies, the particular firing conditions during a time frame decades in the past may be unknown and irrecoverable. If irreducible errors dominate overall uncertainty in sediment-penetration prediction, then there is little overall fidelity to be gained by improving the reducible uncertainties.

Continuing this analysis, we will move from a consideration of the error in one instance (e.g., one munition impacting the sediment) to the statistical characterization of the error over a large number of instances (e.g., a large number of munitions impacting the sediment). We denote the standard deviation of the error ; this parameter represents the expected or mean magnitude of the corresponding single-instance error over a large number of instances.

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Improving Initial Burial Prediction

Improving model ( , ) vs. improving measurement ( ) Plan:

Evaluate model sensitivities by running models and using low-orderanalytic approximations

Get waterline input errors , , from ballistics community Get and from oceanography community

Only measurement contribution to can be improved Review validation experiments for hydrodynamic models to estimate

model uncertainties ( , ) May be different in high-speed regime

Plug everything into sensitivities equation:

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One key question for efficient improvement of a sediment-penetration model is the choice between model improvement and measurement improvement:

Model improvements could help if the aspects of the physics not captured or poorly captured by the model contribute significantly to the deviation of predicted depths from actual penetration depths.

On the other hand, if the physics is already adequately captured by the model but there is sensitivity to knowable inputs, measurement improvements could help.

In short, if the dominant uncertainty is due to unknowable inputs, other improvements may have little leverage.

Here we lay out a recipe for considering the relative error contributions for cases of interest. We have exercised the first and last steps for an example case.

We hypothesize that although low-order models may not capture the nuances of the physical world, they do capture the gross behavior of the physical world well enough to identify major sensitivities. These low-order models are well-suited to use in this fashion due to their quick run time and the insight provided by analytic approximations. We derive and evaluate the sensitivities in an example to follow:

The initial conditions at the waterline ( , , ) should be provided by the ballistics community, where as a matter of course in the testing and characterizing of munitions under nominal conditions, there should be an understanding of the scatter in their speed, trajectory, and angle of attack.

Variability and measurement uncertainty in the environmental parameters ( and ) should be drawn from the oceanographic community regarding water depth (tides, waves, and long-term water-level variation) and sediment properties (composition, spatial variability, variation over time, and measurement errors). In general, only the variance in measurement error of sediment properties will be reducible.

The model uncertainties should be estimated using existing validation experiments where possible. Prediction errors of the models compared with the experiments should provide the scope of the magnitude of the modeling error for the regime tested. Care must be taken to ensure the free model parameters were not set using the same experiments considered for validation. If experiments were only done in the low-speed regime, as is the case for the mine burial models, new experimentation may be required to understand model errors in the high-speed regime. In particular, the use of low-speed drag and moment coefficients may not capture the high-speed hydrodynamic behavior of munition projectiles.

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Finally, as we show in our example, the error contribution estimates should be evaluated in the overall sensitivity equation to compare their magnitudes and identify potential opportunities for improvement.

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Sensitivity Equation at Mudline:

Assuming low cross-correlation contribution:

Error Budget: Sediment Penetration Model

Errors emerging from hydrodynamic calculation including hydrodynamic modeling error and input error to hydrodynamic model , ,

Sediment model sensitivities Sediment Modeling Error

Sediment property uncertainty

What would each of these terms be? Depends on munitions, sediments…

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Looking at the sediment-penetration model, we make the conversion from single-instance errors to standard deviations to form what we call “the sensitivity equation” for the sediment phase. In general, there could be cross-terms in the sensitivity equation involving correlations between the various error sources. For instance, deviations of velocity from the nominal value may be correlated with deviations from the nominal trajectory. For simplicity, we assume that such cross-terms are negligible here. For truly unrelated terms, such as the sediment properties and the munition projectile’s initial conditions at the mudline, this assumption is strongly justified. Regarding the munition projectile’s angle of attack, symmetry might suggest that the cross-terms with the other initial conditions would be zero (e.g., if for every there is equal likelihood of and then, on average, the product will equal zero because the positive and negative contributions will cancel out by symmetry).

The sensitivities and initial condition uncertainties depend on the so-called operating point—the particular nominal initial conditions and munitions properties. The sediment property uncertainty likewise depends on the sediment properties themselves.

Note that in writing the error budget for the sediment-penetration phase, the errors in the initial conditions at the mudline implicitly depend on uncertainties propagated from the waterline initial conditions via the hydrodynamic model.

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Waterline

Mudline

h

1. 2D hydrodynamic model:

2. Aubeny’s sediment penetration model1

Modeling assumptions:• Presented area and drag coefficientnumbers are comparable with thoseof Strike35 (i.e., munition tumblesupon impact)

• Broadside munition sediment impact• Known sediment shear strengthdistributions

Stages of a Simplified UXO Burial Depth Model

Δ2

2

12 , 2

12 , 2

1 ,

,

where: is the force experienced by the munition,

, , , , are empirical constants

1 C. Aubeny and H. Shi, “Effect of Rate‐Dependent Soil Strength on Cylinders Penetrating Into Soft Clay,” IEEE J. Oceanic Eng. 32 (1) (January 2007).

89

To gain understanding of how a munition projectile propagates through the hydrodynamic and sediment phases, we make simplifying assumptions and solve the resulting equations of motion numerically in 2D. In our formulation, “z” and “x” are the vertical and horizontal directions, respectively.

Hydrodynamic Phase: The differential equations describing the trajectory of the moving munition are given on the chart in part 1 of this slide. The unknowns in these equations are presented area and drag coefficient. We estimate the presented area by assuming that the munition tumbles upon impact. We estimate the value of the drag coefficient by analyzing results of the STRIKE35, as shown in the next chart.

Sediment Phase: We use Aubeny and Shi’s (2007) equations of a projectile propagating through the sediment phase, given in this chart in part 2. We augment this model by adding a sediment inertial resistance force term. We use the numerical values for the model parameters given by Aubeny and Shi (2007): f1 =7.41, f2 = 6.34, b1 = .37, b2 = .155,=.022. Note that when the sediment-impact velocity is small, on the order of several meters per second, the inertial force effects are also small and can be neglected. This does not remain so, however, as the impact velocity increases.

90

Example: M344m = 8.75 kg, 0.106m, 0.57m, 0.3, = 200 m/s, = 30°, = 0

Simplified 2D Hydrodynamic Model vs. IMPACT35 (3D)

It is possible to use simplified 2D hydrodynamic models in place of IMPACT35 (3D).

This simplified model was solved numerically.

0 2 4 6

2

4

6

8

Impact 35IDA model

horizontal distance, mU

XO

dep

th, m

0 1 2 3

2

4

6

8

Impact 35IDA model

time, sec

UX

O d

epth

, m

Presented area 0.045 m2

91

Hydrodynamic Phase Here we numerically solve the 2D hydrodynamic phase equations of motion for a particular case with the impact conditions and

munitions parameters as shown. The trajectories predicted by our simplified 2D model (dashed gray line) and STRIKE35 (solid black line) are in good agreement, lending confidence to using our simplified 2D model for subsequent analyses of munition trajectory through the water.

92

Even Simpler 1D Hydrodynamic Model

An analytic model is an exact solution to a simplified problem andcan be used to: Compute partial derivatives (for the sensitivities equation). Scope the area of parameter space in which some variables do

not contribute (e.g., terminal velocity).

In a regime far above terminal velocity in water, the forces on thedescending munition will be dominated by hydrodynamic form drag:

where

≡

93

Hydrodynamic Phase In contrast to computational models (which are in some sense like a black box, a mysterious enclosed device, where one can explore

behavior by adjusting inputs and observing outputs), analytical models provide a mathematical formula, that can be examined to produce insight. For instance, as shown on the bottom of this slide, the analytical solution to a hydrodynamic model, , suggests that projectile velocity at the mudline will be proportional to projectile velocity at the waterline (with a constant of proportionality that will vary with projectile properties). This solution may be inaccurate because it solves a simplified problem stripped of some of the important physics such as projectile rotation, but assuming it captures the dominant physical features, the insights it yields should hold approximately true. Thus, without doing many computational experiments, one could understand how projectile velocity at the mudline depends on other factors.

This analytical model can be used to compute mathematical expressions for the sensitivities in the error budget (e.g., ). It can

also be used to carve out areas of parameter space (sets of operating points) where variables may be unimportant. For instance, for sufficiently deep water, munitions will reach terminal velocity (the velocity at which vertical forces are in equilibrium and the object remains at a constant velocity) before hitting the mudline. In such water depths, the munition’s velocity at the waterline no longer affects its velocity at the mudline, and improved hydrodynamic models to capture detailed projectile dynamics during the early phases of water entry are wasted with regard to sediment-penetration depth prediction. Therefore, the analytic model can help determine how deep is “sufficiently deep” such that the sediment-penetration depth is independent of the initial conditions at the waterline.

The particular model shown on this slide ( ) assumes that the projectile travels only vertically (i.e., in 1D) and presents a constant projected area and drag coefficient. In this case, the projectile velocity decreases exponentially with depth in the water column. In the backup slides (“Angled Trajectories”) we show a more nuanced analytical model that captures horizontal motion of a 2D trajectory.

94

Here we use an even simpler 1D hydrodynamic model in place of IMPACT35.

We solved this model analytically.This model delineates a region of parameter insensitivity

due to terminal velocity (vt).

M344 Specific gravity: S = 4 = 0.3 Aspect ratio = 6 Diameter = 0.106 m Presented area depends

on dynamics Nose on Spinning Broadside

Depth to terminal velocity

Water dep

th,

(m)

(m/s)

1.1

Even Simpler 1D Hydrodynamic Model: Example

Broadside

95

Hydrodynamic Phase As suggested on the previous slide, here we exploit the 1D analytical hydrodynamic model ( ) to identify the depth at

which the munition’s terminal velocity is achieved and its subsequent behavior is independent of the initial conditions (

0). Since, in this model, terminal velocity is approached asymptotically, we consider achievement of 110% of terminal velocity

sufficient to remove the effect of initial conditions. We take a set of munitions parameters associated with the M344 shell. See backup slide “M344” for more information on this particular type of projectile.

We consider three possibilities for the projectile dynamics in the water (parametrically, since the model does not allow projectile rotations):

nose-on,

spinning, and

broadside.

The presented area of a projectile is minimized when traveling nose-on and maximized when traveling broadside, so results for those two conditions should bound the behavior in other cases. For a projectile that spins through the water column, one can show that the analytical solution holds with the presented-area-drag-coefficient product replaced by its time-averaged value,

, .

The analysis then shows that for this projectile, if unstable in the water column (either broadside or spinning), initial conditions at the waterline do not influence sediment-penetration depth in water deeper than 15 m. In contrast, if this projectile transits the water stably nose-on, it would need 30–40 m depth to wash out initial waterline conditions.

96

Hydrodynamic Instability

If fins break off or with a large enough angle of attack at thewaterline ( ), the projectile can tumble or turn broadside. In our computational studies, we utilize a broadside projectile as the

worst case for UXO exposure (i.e., shallowest sediment penetration).

Angle of attack at mudline Stable:

0 Unstable: flips broadside on sediment impact

0 Unstable: random impact

0to2

time

Broadside Flop

97

Hydrodynamic Phase The UXO of greatest concern will be those most shallowly buried. Due to the increased hydrodynamic and sediment-penetration

resistance, these will be munitions that impact the sediment broadside. Munitions that meet the water at a sufficiently high angle of attack or those that break one or more fins are likely to tumble or turn broadside in the water. Therefore, for our computational studies, except where otherwise noted, we examined munitions traveling broadside through the water.

We considered three possibilities for munitions behavior at the water-sediment interface:

For munitions moving stably nose-on through the water, we presume they maintain a low angle of attack in a narrow distribution with a mean of zero (perfectly nose-on) until mudline contact.

For spinning munitions, the angle of attack at the mudline will be random—for these munitions, the differential formulation of the sensitivity (with partial derivatives) will not accurately capture the effect and finite differences should be considered.

For angle of attack sufficiently deviating from nose-on, the munition is likely to flip broadside to the sediment interface due to the torque exerted by the sediment resistance at the munition’s point of first contact.

Again, to consider the worst case for UXO risk, we assume the last case, what we call “the broadside flop,” unless otherwise stated in our analysis. The broadside flop also fixes 0 , so sensitivity to is irrelevant (it gets multiplied by zero in the sensitivity equation).

98

Simplified Sediment Model

Burial depth estimate from simplified sediment model (Aubeny and Shi 20071):

sin1

2 sin cos

Sensitivities of penetration depth estimate to initial conditions at mudline:

21

cot

Sensitivity Equation:

11

cos sinsin cos

11

Now plug these derivatives back into the Sensitivity Equation…

Details available in backup

1C. Aubeny and H. Shi, “Effect of Rate‐Dependent Soil Strength on Cylinders Penetrating Into Soft Clay,” IEEE J. Oceanic Eng. 32 (1) (Jan. 2007).

99

Sediment Phase To evaluate the sensitivities in the sensitivity equation for the sediment phase, we take the velocity-dependent resistance model

published by Aubeny and Shi (2007). The velocity (i.e., rate dependence) prevents derivation of an analytic solution to Aubeny and Shi’s model. However, this rate dependence is weak, and by considering the range of the rate-dependent contribution, bounding analytic solutions (upper and lower bounds to the penetration predicted by the full rate-dependent Aubeny and Shi model) can be derived as we do in the backup slides (see “Simplified Sediment Model”). The form of the solution is shown on the current slide. This expression is amenable to taking derivatives. We do so and utilize the resulting expressions in the sensitivity equation.

100


With derivatives plugged in…

21 cot

11

cos sinsin cos

11

Sensitivity equation:

For an unstable, broadside flop: By definition, we know α and so = 0

time

42

Therefore:

101

Sediment Phase To evaluate the sensitivities in the sensitivity equation for the sediment phase, we take the velocity-dependent resistance model

published by Aubeny and Shi (2007). The velocity (i.e., rate dependence) prevents derivation of an analytic solution to Aubeny and Shi’s model. However, this rate dependence is weak, and by considering the range of the rate-dependent contribution, bounding analytic solutions (upper and lower bounds to the penetration predicted by the full rate-dependent Aubeny and Shi model) can be derived as we do in the backup slides (see “Simplified Sediment Model”). The form of the solution is shown on the current slide. This expression is amenable to taking derivatives. We do so and utilize the resulting expressions in the sensitivity equation.

102

Use computational hydrodynamic model to evaluate and :

Estimating Errors in Initial Conditions at Mudline

Monte Carlo, ,

and likewise for

E.g., compute as a function of ,holding and constant:

0 20 40 60 80 10020

40

60

80

100

h = 5 m= 300 m/s

h = 5 m= 50 m/s

h = 10 m= 50 m/s

h = 10 m= 300 m/s

103

Combine Hydrodynamic and Sediment Phases This slide shows the sediment impact angle as a function of water impact angle for two water depths : 5 and 10 m, and

two water impact velocities : 50 and 300 m/s. As can be seen from the figure, the sediment impact angle varies from about 35 degrees (when the water depth is lowest, water impact velocity is highest, and water impact angle is about 30 degrees) to 90 degrees (when the water impact angle is also 90 degrees).

We proceed to utilize our simplified 2D hydrodynamic model to estimate distributions of the sediment impact angle and velocity by performing Monte Carlo simulations for various water depths and water impact angle and velocity values.

104

uniformly distributed, 50 to 300 m/snormally distributed, 45±15°

Shallow water:Uniformly distributed 1–10 m

4.88 4.9 4.92 4.94 4.960

20

40

60

80

100

sediment impact velocity, m/s

UX

O fr

actio

n

85 86 87 88 89 900

20

40

60

sediment impact angle, degreesU

XO

frac

tion

0 50 100 1500

0.2

0.4

0.6


UX

O fr

actio

n

25 27m/s

20 40 60 80 1000

0.05

0.1

sediment impact angle, degrees

UX

O fr

actio

n

56° 13°

Monte Carlo Results: Estimates of and

Deep water:Uniformly distributed 15–30 m

105

Combine Hydrodynamic and Sediment Phases To account for unknown initial conditions of munitions impacting the water surface, we assume that water depth , water impact

angle , and water impact velocity can be represented by probability distributions with parameters shown on this slide. We distinguish two fundamentally different cases: shallow water (water depth varies uniformly from 1 to 10 m) and deep water (uniform water depth variation from 15 to 30 m):

In deep water (bottom row), the sediment impact velocity centers around its terminal velocity value with very a small deviation, while the sediment impact angle is very close to 90 degrees. The simulations show at that at a water depth of 15 m and higher, the effect of the waterline impact conditions is practically negligible, and the sediment impact can be approximated by the munition’s terminal velocity with vertical impact angle.

In shallow water (top row), both the sediment impact velocity and angle are widely distributed with the mean and standard deviations shown on the figure. Clearly, in this case, the sediment-penetration depth predictions will be much more uncertain than those for the case of deep water.

106

Operating Point and Initial Condition UncertaintyCases

m/s m/s

Shallow 25 27 56° 13° 50°0 50%*

Deep 4.9 0.5 70° 0.7° 90°Contribution to Initial Burial Depth Uncertainty TotalShallow 216% 15% 0

50%222%

Deep 20% 0% 0 54%

Total square error contribution

DeepShallow

Our Example: Unstable, Broadside Flop

42

For the Unstable, Broadside Flop case:

• In shallow water: Uncertainty in dominates the uncertainty in

• In deep water: Uncertainty in dominates the uncertainty in

e.g., 27 m/s error in leads to a 216% error in

*Rennie and Brandt, “SERDP Project MR‐2227 Interim Report,” August 2015.

107

Combine Hydrodynamic and Sediment Phases Using results from the Monte Carlo analysis and the scientific literature to evaluate the operating points and the variance of the

mudline initial conditions (only the contribution from initial waterline condition variation), we evaluate the sensitivity equation and compare the error contributions:

In shallow water, the waterline projectile velocity significantly influences the mudline projectile velocity , so variation in sediment impact velocity is not erased by water depth . Uncertainty in the projectile velocity at the mudline then dominates the uncertainty in the sediment-penetration depth . Improving the penetration model would only help if the fractional model error / exceeds 100%. Since the velocity uncertainty considered here, , ultimately stems from the munition firing condition variation, it is irreducible. Additional improvements in measurement or modeling will only be useful if their current contribution exceeds a 100% fractional error.

In deep water, where the effect of the irreducible uncertainties of the firing conditions are effaced by the achievement of terminal velocity, uncertainty in the sediment properties dominates the overall sediment-penetration prediction uncertainty

. If the uncertainty in sediment properties stems from measurement error, then better measurement techniques might improve sediment-penetration prediction accuracy. On the other hand, if is dominated by point-to-point spatial variation in the sediment properties over the local region of interest, then will be irreducible, and further model or measurement enhancements will do little to improve the accuracy of sediment-penetration prediction.

108

Operating Point and Initial Condition UncertaintyCases

m/s m/s

Shallow 25 27 56° 13° 50°0 50%*

Deep 4.9 0.5 70° 0.7° 90°Contribution to Initial Burial Depth Uncertainty TotalShallow 216% 15% 0

50%222%

Deep 20% 0% 0 54%

Total square error contribution

DeepShallow

Our Example: Unstable, Broadside Flop

42

Recommendation #9:Use simplified models with sensitivity analytic framework to understand when and how

initial sediment penetration predictions can be improved.

e.g. 27 m/s error in leads to a 216% error in

109

Combine Hydrodynamic and Sediment Phases As we have illustrated by this example, the analytical models can be a powerful tool to reveal opportunities (or lack thereof) for

improving sediment-penetration prediction. Higher fidelity models, where they exist and are sufficiently validated, can also be used to evaluate the sensitivity equation, where feasible.

110

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

burial depth, m

UX

O fr

actio

n

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

burial depth, m

UX

O fr

actio

n

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

burial depth, m

UX

O fr

actio

n

uniformly distributed, 50 to 300 m/snormally distributed, 45±15°

fixed and known

Shallow water1–10 m

Deep water15–30 m

0.0598 0.06 0.06020.06040.06060.0608 0.0610

0.1

0.2

0.3

= 20 kPaSand

,

0 0.5 1 1.5 20

0.2

0.4

0.6

burial depth, m

UX

O fr

actio

n = 4 kPaSilty clay

Irreducible Uncertainty in Burial Depth

In shallow water, initial conditions greatly affect the uncertainty in sediment penetration depth.

In deep water, the effect of initial conditions on sediment penetration depth is negligible.

0.0186 0.0187 0.0188 0.0189 0.0190

0.2

0.4

0.6

0.8

,

, ,

Sand Silty clay

111

Combine Hydrodynamic and Sediment Phases Having estimated variability in sediment-impact conditions, we proceed to estimate variability of predicted sediment-penetration

depths. As an example, we consider two sediments representative of low sediment shear strength (silty clay) vs. high sediment shear strength (sand).

As we have shown on the previous slide, there are two fundamentally different regimes that we have to consider: deep water and shallow water:

In deep water, the water impact conditions have negligible effect on the predicted sediment-penetration depth ; they are instead controlled by the munition terminal velocity and sediment shear strength .

This is not so for the shallow case: the sediment-penetration depth is distributed from near zero to 0.4 m values in sand and up to 1.5 m in silty clay. Note that in the shallow-water case, these resulting depth distributions ultimately arise from the unknown water impact conditions, and the predicted depth uncertainty is therefore irreducible.

112

WHERE WOULD ADDITIONALFIDELITY HELP?

113

We have now explored:

What fidelity (i.e., accuracy and precision) is useful for underwater UXO remediation projects when estimating a munition’s initial penetration depth into the sediment.

What fidelity is already achievable via existing penetration models that have already been developed for underwater mines.

We will now consider our third and final question: Where (and when) would additional fidelity be helpful for underwater UXO remediation projects?

114

• Improving the model will not fully eliminateuncertainty in initial sediment penetration depth

due to irreducible uncertainty inenvironmental conditions ( ) and initial impactconditions at the waterline (v00, etc.).


0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1Cumulative distribution of UXO burial depth in sand

burial depth, m

prob

abili

ty

Fraction of potentially mobile UXO at the time of impact

Mob

ility limit

Burial depth , m2

Connecting Useful Fidelity to Achievable Fidelity:Shallow Water Example

115

Consider, as an example, a particular case of munitions buried in sand in shallow water depths, 3–5 m. In the upper right figure, we show the cumulative distribution of the buried depth, in this case based on the uncertainties in:

the waterline impact velocity (we assume a uniform distribution, 50 to 300 m/s),

the waterline impact entrance angle (we assume a normal distribution with mean = /4 and standard deviation = /12) and,

the sediment shear strength (we assume a normal distributed with mean = 20 MPa and standard deviation = 5 MPa).

As can be seen from the upper right figure, slightly more than 40% of munitions will have a sediment-penetration depth less than 1/2 the diameter D, making them potentially mobile. In other words, our simplified models predict that about 40% of the munitions will be initially mobile.

Note that the spread of the cumulative distribution in the upper right figure stems from the uncertainty in the waterline initial conditions and the sediment shear strength and is therefore irreducible. We can overlay this irreducible uncertainty on the burial regime map developed in the first part of our analysis, as we show in the bottom left corner by a blue arrow. (The magnitude of is estimated from the cumulative distribution to be about 1.5 times the munition diameter D.) If the munition lands in an map regime where the bottom water velocity is not high enough to make the munition mobile, then the munition will remain stable. Improving the accuracy of the model (either in the hydrodynamic or sediment phases or in both) will not lead to improving our knowledge of the fate of this munition, given this large (irreducible) uncertainty in its initial sediment-penetration depth .

116


due to irreducible uncertainty inenvironmental conditions ( ) and initial impactconditions at the waterline (v00, etc.).


0 0.1 0.2 0.30

0.2

0.4

0.6

0.8


burial depth, m

prob

abili

ty


Mob

ility limit

Burial depth , m

ErosionMudline

Annual variation of sediment level*

2


117

Here we show the effect of sediment floor height variation on the munition burial depth . Consider, as an example, a particular case of munitions buried at a specific location: Duck, North Carolina. Shown on the upper left corner is Duck’s reported yearly variation of sediment height, quantified as:

the water depth versus distance from shore (brown curve, vertical axis on left) and

the variation in the water depth versus distance from shore (red curve, vertical axis on right).

In shallow water depths (i.e., about = 5 m), the variation in water depth is about = 0.2 m. One can think of this variation as either the variation in the water depth or as the variation in the sediment level.

We ask the following question: How will this variation in sediment level, , affect the uncertainty in the penetration depth of munitions, ?

118


due to irreducible uncertainty inenvironmental conditions ( ) and initialconditions at the waterline (v00, etc.).

• Sediment floor erosion/accretion will wipe outthe effect of uncertainty in initial penetrationdepth .


0 0.1 0.2 0.30

0.2

0.4

0.6

0.8


burial depth, m

prob

abili

ty

New fraction of UXO potentially mobilized by erosion


Mob

ility limit

Burial depth , m

ErosionMudline


2


0.2m

119

If the local bottom floor conditions are such that these munitions are initially immobile, the scour process and local sediment floor variation (i.e., accretion) may bury the munition, making the magnitude of initial sediment-penetration depth irrelevant. On the other hand, the sediment-level variation can also expose the munition and make it potentially mobile if it takes place on the timescale shorter than that of the scour process. For example, at a water depth of about = 5 m, a sediment-level variation of about = 0.2 m may expose initially buried munitions and increase the percentage of the munitions that are mobile (from just over 40% to about 97%, as shown in the upper right figure).

We use a red arrow to represent the sediment-level variation in the burial regime map in the bottom left. We observe that, in addition to the uncertainty in initial burial depth , the sediment-level variation contributes irreducible uncertainty in increasing (or decreasing) the munition penetration depth . Improving the model accuracy will unlikely be useful in this case.

This example demonstrates that under certain conditions, the depth of the munition’s initial penetration into the sediment does not affect its subsequent exposure and mobility; in this case, an improvement in the accuracy of the munition’s initial penetration prediction does not lead to more accurate assessment of the munition’s fate.

120



• Sediment floor erosion/accretion will wipe outthe effect of uncertainty in initial penetrationdepth .

• The only risk bucket sensitive to model fidelity isinitial mobility.


0 0.1 0.2 0.30

0.2

0.4

0.6

0.8


burial depth, m

prob

abili

ty



Mob

ility limit

Burial depth , m

ErosionMudline


2


0.2m

121

The burial regime maps can help to answer the question: When will additional improvements in model accuracy be useful?

If the bottom water velocity is low and the munition remains immobile after the impact, then the scour processes and sediment floor depth variations will control the fate of the munition and erase the effect of the initial sediment-penetration depth, reducing the usefulness of the model fidelity improvements.

If, on the other hand, the bottom water velocity is such that the munition can be mobile, then the munition’s mobility is determined by its initial sediment-penetration depth, and the accuracy of the penetration predictions is very important because it will influence the mobility and fate of the munition.

These considerations show that usefulness of improving model accuracy depends on the particular case of interest. In our example, accurately predicting the initial mobility of munitions may help estimate the fraction of potentially mobile munitions, even though there is considerable irreducible uncertainty in their initial sediment-penetration depths.

122



• Sediment floor erosion/accretion will wipe outthe effect of uncertainty in initial penetration depth .

• The only risk bucket sensitive to model fidelity isinitial mobility.


0 0.1 0.2 0.30

0.2

0.4

0.6

0.8


burial depth, m

prob

abili

ty



Mob

ility limit

Burial depth , m

ErosionMudline


2


0.2m

Recommendation #10:Use regime map to evaluate whether proposed improvements in model fidelity will have

operational utility.

123

Thus, our final recommendation is to use a burial regime map to evaluate whether proposed improvements in model fidelity will have operational utility.

124

FINDINGS AND RECOMMENDATIONS

125

We now summarize our findings and recap our recommendations.

126

Findings

In deep water, initial conditions at the waterline , , α ) do notinfluence the initial sediment penetration depth ( ). Terminal velocity ( ) is the dominant influence on the initial conditions at

the mudline ( , , α ), which, in turn, influence the munition’s initialsediment penetration depth ( ).

, and therefore , is sensitive to the hydrodynamic model and, inparticular, the munition’s drag coefficient.

In shallow water, much of the uncertainty in the munition’s initialsediment penetration depth ( ) stems from the variability in initialconditions at the waterline ( , , α ). This variability is based on the original firing conditions and is irreducible.

127

Findings

Sediment shear strength ( ) significantly influences the munition’s initial sediment penetration depth ). Geographic variability in within a region of interest produces irreducible

uncertainty in , putting a lower bound on the achievable precision in . In deep water, where is low, has strong influence on .

Erosion/accretion strongly affects exposure/mobility. Uncertainty in puts an upper bound on the useful fidelity in .

A simple mathematical framework (i.e., “the sensitivity equation”) coupled with simple analytical models can be applied to specific cases to compare the present-day, achievable model errors to what is useful.

128

Recommendations

1. Improve sandy sediment-penetration models.2. Develop mobility models for silt.3. Develop mobility models for partly buried objects.4. Develop scour burial models for silt.5. Improve models of consolidation and creep.6. Exercise caution in improving hydrodynamic models to support initial sediment-penetration

estimates—the effect of better hydrodynamics may be dwarfed by stochasticity due tounknown precise initial conditions at the waterline.

7. Existing sediment-penetration models (other than STRIKE35) are designed for near-cylindrical mines—for munitions, however, projectile-specific drag, lift, and momentcoefficients are needed for estimating hydrodynamic stability and gross velocity.

8. Modules of existing depth penetration models are nearly independent and could andshould be mixed and matched with little effort to choose best-of-breed for each phase(aero/hydrodynamic/sediment).

9. Use simplified models within the sensitivity analytical framework to understand when andhow initial sediment-penetration predictions can be improved.

10. Use burial regime map to evaluate whether proposed improvements in model fidelity willhave operational utility.

129

Backups

130

/22 1 sin

sin 1 2

Equilibrium of moments:

Δ 2 cos14 2 1 sin

Δ

1

1

1

Expand existing mobility model to include partial burial

Mobility Model: Balance of Moments about Contact Point

131

IDA developed our own model for predicting mobility thresholds, consistent with the phenomenological model quoted in section 4.2 of Rennie and Brandt (2015) for fully unburied munitions but extending to partly buried munitions via physics first principles.

132


Used the Bearing Factor Model (from IMPACT28 and IMPACT35) Net retarding force =

Buoyancy + Hydrodynamics + Sediment shear resistance + Inertial drag For dominant sediment shear resistance:

1 ln , where sin cos

Based on data from Aubeny, the term in parentheses 1 lnranges in value from 1 to 1.5.

Therefore, we can bound the solution by setting that term to aconstant: 1 ln , where 1, 1.5:

Integrating the equation once:

And therefore,

133

This slide describes IDA’s simplified sediment model, adapted from Aubeny and Shi (2007).

To do the first integration of the resulting simplified differential equation , we first multiply both sides by :

Noting that , it becomes apparent that both sides of this nonlinear equation are exact time differentials. Integrating both sides with respect to time and applying the initial condition that when 0, then yields:

12

12 1

or

21 .

The initial burial depth is the vertical projection of the penetration distance at the time when the velocity goes to zero:

sin1

2 .

In the Aubeny and Shi model, the exponent changes when the object is half-buried, so for burial deeper than , one would need toapply a two-step solution, first solving for the velocity at half-burial and then treating that as the initial condition for further burial. In our computations, we used a single set of Aubeny constants equivalent to those used in IMPACT35.

For b = 0.155 (Aubeny and Shi’s value for more than half-buried cylinders), the bounding estimates for differ by 30% (i.e., for 1.5, the initial burial depth will be 70% of the initial burial depth for 1, and the model with continuously varying shear strength

based on instantaneous velocity will give an intermediate result).

At high sediment-impact velocities, inertial drag (form drag or dynamic pressure) becomes a significant contributor to dynamics and should not be ignored. Inertial drag is dominant when

≫ .

134

For example, for a drag coefficient of ~1/3, sediment density of 3000 kg/m3, ~10 , ~10, and ~1, inertial forces would dominate for ≫ 10 m/s. In this region, the velocity will exponentially decay with depth in the same fashion as discussed in the hydrodynamic terminal velocity model on backup slide “1D Projectile Dynamics”:

.

136

Penetration Depth in Sandy and Silty Clay: Deep Water

Broadside impact:• uniformly distributed, 50 to 300 m/s• normally distributed, 45±15°

0 0.1 0.2 0.30

0.05

0.1

0.15

0.2

burial depth, m

UX

O fr

actio

n

• In sand, all UXO may be mobile, hence scour models are needed in low bottom water velocity areas; mobilitymodels are needed in high bottom water velocity areas; sediment penetration at impact is important tounderstanding in which regime the UXO is; therefore, accurate models for penetration into sand at velocity closeto terminal are useful.

• In silty clays, knowledge of shear strength, as well as accurate scour, mobility, and long‐term sedimentpenetration models, are important for assessing the fate of UXO.

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

h = 15m, sandh = 15m, silty claymobility limit

Cumulative distribution of UXO burial depth in Silty Clay and Sand

burial depth, m

prob

abili

ty

Silty Clay

exposed

137

This slide demonstrates selected results regarding the deep-water case:

In sandy areas, most if not all munitions are exposed and potentially mobile. When the sediment level variations take place on the timescale longer than that of scour, a munition’s equilibrium scour burial depth will be determined by the scour process. A model with an accurate description of these phenomena is needed.

In silty clay areas, fewer than half of all munitions will be immobile after sediment impact. In this case, more accurate measurements of silty clay shear strength, as well as models of the scour process and long-term creep and consolidation processes, are needed for more accurate estimates of the munition’s fate.

138

Conservation of momentum:

12 Δ

Terminal velocity: Net force = 0

∗ 2Δ

Exact solution:

∗ 1 ∗ 1 ∗

∗ ≡2

Combined weight and buoyancy

Drag

water densityΔ projectile density ‐

projectile massprojectile cross‐sectional areadrag coefficientprojectile volumegravitational accelerationprojectile velocity at waterline

buoyancy

weightdrag

1D Projectile Dynamics

139

When inertial drag dominates gravitational forces (buoyancy and weight) and viscous forces, 12 .

This differential equation can be separated to form

2 ,

which can be integrated to form 1

21 .

Integrating once more to find the depth z as a function of time, 2

ln 21 2

ln .

This demonstrates exponential decay of velocity with depth,

.

Even at a velocity of 1 m/s, for a 10 cm diameter munition, the Reynolds number of flow through the water would be on the order of

~10 ,

so inertial drag is expected to dominate viscous drag.

Terminal velocity occurs when the resistive drag forces balance the driving gravitational forces:

140

12

∗ Δ .

Solving for the terminal velocity ∗ yields

∗ 2Δ.

For a shape approximating a cylinder, ℓ, cos ℓ sin . Thus, the terminal velocity scaling with munition size is revealed as

∗ 2 Δ ℓ, .

In the exponential-decay model, this velocity is reached after a finite period of time at a depth

∗ 2ln ∗ .

Let us denote the length scale L∗ ≡ .

In a more nuanced model where buoyancy and weight act throughout, terminal velocity is only approached asymptotically. Solving the full conservation of momentum equation

12 Δ

or

∗ ∗

which can be separated to form

141

∗ ∗

and then integrated to yield

∗ coth∗

∗ tanh∗.

This can be integrated again to yield

∗ ln sinh∗

∗ tanh∗

ln ∗ 1,

which with the initial conditions becomes

∗ 1 ∗ 1 ∗.

This expression only allows for asymptotic approach to the terminal velocity, as predicted. To compare this model to the previous inertia-only forces with finite time to terminal velocity, let us substitute the expression for the depth to terminal velocity in the prior model and compute the velocity achieved at the same depth in the more accurate model:

∗ 2∗.

In the limit of high initial velocities, the inertia-only model predicts terminal velocity achievement at a depth where the velocity would actually be 40% higher.

142

∗ 2Δ∗ 2

water densityΔ projectile density ‐

projectile massprojectile cross‐sectional areadrag coefficientprojectile volumegravitational accelerationprojectile velocity at waterline

Depth to Terminal Velocity

143

For projectiles traveling straight downward, here we show, relative to the scale length ∗, how deep the projectile would have to penetrate the water to achieve come within 10% or 1% of terminal velocity as a function of its initial velocity at the waterline (as a multiple of terminal velocity).

How can the drag coefficient be treated for a projectile spinning during its descent? In the inertia-only model, the drag coefficient can be represented as a function of time ,

12 .

Integrating the separated equation now takes a slightly different form:

21 1

.

For a projectile experiencing many cycles of rotation, the integrated drag coefficient can be taken as a time-invariant average multiplied by the total time, . The derivation then continues as before with replaced by . Thus, the drag coefficient can just be treated as its time averaged value in the final expressions.

STRIKE35 employs an expression for the drag-coefficient-area product (treated as drag coefficient variation with a constant nominal presented area) for the Joint Direct-Attack Munition (JDAM), whose dependence on angle of attack is given by a coefficient

equal to (Chu et al. 2011). For a projectile with a similar angle-of-attack dependence, spinning would sample all angles of attack equally leading to

2 1√2

erf√2

0.4.

Thus, for this type of angle of attack dependence, the effective drag coefficient of a spinning projectile will be 40% of its broadside value.

144

Only a component of gravity cos counteracts drag.The remainder moves the trajectory angle closer tovertical.

Therefore, the path length to terminal velocity is lessthan the depth to terminal velocity for a vertical velocity.

Depth to terminal velocity can be bounded:∗ ℓ∗ sin

12

Δ ∗

where ℓ∗ is the path length to terminal velocity in the vertical trajectory case.

Angled Trajectories

145

If the trajectory of the projectile is not straight downward, at every point along the trajectory, gravity acts with a component along the trajectory counteracting drag and a component perpendicular to the trajectory acting to turn the trajectory further downward. The action of the gravitational forces along the trajectory will therefore always be less than for a straight downward trajectory as considered in the full model on p. 138 and more than in the model discussed in the notes from the same chart ignoring gravitational forces entirely.

Let ℓ∗ be the depth to terminal velocity without gravitational forces in the straight downward case. A lower bound to the depth to terminal velocity will be ∗ ℓ∗ sin . We are more interested in an upper bound to identify regions of phase space where projectile initial conditions at the waterline do not contribute to initial burial depth. A simple upper bound can be derived by assuming that the gravity-free depth (ℓ∗ sin is augmented by the gravitational contribution with no additional drag force due to the associated higher velocity. The vertical travel of sinking projectile absent drag will be given by ∗ (using ∗ from the straight downward case)

leading to the simple but overly loose upper bound

∗ ℓ∗ sin12

Δ ∗ .

For a more sophisticated bound, consider that with a downward curving trajectory, and thus, sin sin . Next consider an uncurving angled trajectory accounting for drag and the tangential component of gravitational forces. Let us call the associated drag forces t . The real trajectory would curve downward leading to larger gravitational components in the along-trajectory direction, larger velocities, and therefore larger drag forces, . Consider the conservation of vertical momentum equation:

sinΔ

sinΔ.

In other words, the right-hand side of the approximation is less negative than the exact right-hand side, so in the approximation, the projectile will decelerate more gradually and reach terminal velocity at a deeper point making it a valid upper bound.

comes from a solution to the along-trajectory momentum equation with a fixed trajectory:

12

Δsin ,

146

the solution to which is found on p. 141 to be

∗ coth∗

∗ tanh∗

where

∗ 2Δ sin ∗ sin .

The drag force is given by 12

12

∗ coth∗

∗ tanh∗.

Substituting this into the bounding differential equation for vertical velocity ∗

∗ coth∗

∗ tanh∗sin

Δ

This can be integrated to yield Δ

cos ∗ coth∗

∗ tanh∗sin

and integrated again to yield

12

Δcos ∗ ln sinh

∗

∗ tanh∗

∗ 1 sin ,

which then gives the depth-to-terminal-velocity-bounded approximation by substitution of ∗ from the straight-downward case (including gravitational forces):

147

∗∗

∗ coth 1 coth ∗ ,

where is the proximity to true terminal velocity considered sufficient in the asymptotic approach: 1 ∗ ∗.

148

M344

JCAMMO.com

149

This slides describes the properties of the M344. With fins broken off, we treated the total length as 57 cm, divided equally between a cylindrical and conical section.

150

Definitions: , : from ballistic/aerodynamic uncertainty : from ballistic/aerodynamic uncertainty + uncertainty in slope of

waterline (waves) :

Short term: waves, tide Long term: water level, mudline evolution

: measurement uncertainty, spatial variation, time evolution

Assumptions: Errors are uncorrelated except maybe and Errors are unbiased except

Flight-path angle transformation to elevation vs. waterline may be biased byobscuration of the back side of waves

Errors in models ( , ) can include unmodeled parameters like rate-dependent sediment strength

Errors: Definitions and Assumptions

151

These next few slides describe our use of the sensitivity equation.

152

Assuming low cross-correlation between terms:

Error Budget: The Sensitivity Equation

153

The Sensitivity Equation at the mudline:

Aubeny and Shi’s sediment penetration model:

sin1

2 sin cos

Deriving sensitivity terms from Aubeny and Shi’s model:

cot

Error Budget: Deriving Sensitivity Terms

Stable case:11

Unstable broadside flop case:0

Unstable random case( between , / ):

Δ1

154

The sensitivity equation at the mudline:

Plugging in sensitivity terms from Aubeny and Shi’s model (stable case):211

cot11

111

where 0.155, so prefactors are 1 , and cot , and so:

42

Example: 10% velocity errors. 50% shear strength errors. Trajectories within 30 degrees of vertical plus or minus 15 degrees. Angle of attack errors less than 10 degrees. With aspect ratios > 5:1. Shear strength errors overall dominant among sensitivities, but if model had 100% error, model error

would dominate.

Error Budget: Comparative Contributions: stable case

155

The sensitivity equation at the mudline:

Plugging in sensitivity terms from Aubeny and Shi’s model (unstable case):211 1

11

where 0.155, so prefactors are 1 , and cot , and so:

42

Error Budget: Comparative Contributions: unstable case

0

156

Sediment-Penetration Model

Sediment shear strength

0 20 40 60 80 1000

1

2

320 KPa10 KPa5 KPa


buria

l dep

th, m

157

This chart shows munition burial depth as a function of sediment impact velocity and shear strength. We utilized Aubeny and Shi’s sediment-penetration model with the addition of inertial terms, as we described in the main text. The munition in this case has a mass of 20 kg and 0.045m2 presented area.

158

Impact angle is normally distributed with mean = 450, std. dev. σ = 150; drag coefficient = 0.3;

0 5 10 15 200

0.2

0.4

0.6

0.8

1

5 m depth10 m depth

100 m/s water impact velocity

bottom impact velocity, m/sU

XO

frac

tion

4 6 8 10 120

0.2

0.4

0.6

0.8

1

5 m depth10 m depth


bottom impact velocity, m/s

UX

O fr

actio

n

0 10 20 30 400

0.2

0.4

0.6

0.8

1

5 m depth10 m depth



UX

O fr

actio

n

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

5 m depth10 m depth



UX

O fr

actio

n

When the water depth is h = 10 m:• the projectile vertical velocity atthe mudline deviates onlyslightly from the terminal velocityof 4.9 m/s

• the largest deviation occurs at = 300 m/s: varies from about 5 to 10 m/s.

In h = 5 m water depth:• when changes from 50 to 300 m/s,the changes from about 1 to 55m/s

• the largest changes are, as expected, at= 300 m/s

Terminal Velocity in Computational Model: Effect of Water Depth and Munition’s Impact Velocity at Waterline on Munition’s Vertical Velocity at Sediment

159

This slide demonstrates the effect of water depth and the munition’s impact velocity at the waterline on the vertical component of the munition’s impact velocity at the sediment:

In 10 m water depth, the vertical component of the munition’s impact velocity at the sediment is very close to the terminal velocity, and the munition’s impact velocity at the waterline has a negligible contribution.

In contrast, in 5 m water depth, the vertical component of the munition’s impact velocity at the sediment is quite sensitive to the variation in its impact velocity at the waterline.

160

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1 to 5 m depth, sand1 to 5 m depth, silty clay1/2 UXO diameter


burial depth, m

prob

abili

ty

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

5 m depth, sand5 m depth, silty clay1/2 UXO diameter


burial depth, m

prob

abili

ty

Fraction of mobile UXO

Fraction of mobile UXO

Selected Results:Fraction of Mobile Munitions in Silty Clay and Sand

161

Under the assumption that the munition is potentially mobile if its sediment-penetration depth is less than one-half its diameter, the cumulative distributions of a munition’s penetration depth can be used to determine the fraction of immobile/potentially mobile munitions, as shown on this slide. In this example, we are considering a uniform distribution of water depths from 1 to 5 m (left) and exactly 5 m depth (right). As can be seen from these figures, the water depth has a larger effect in sand, where the fraction of munitions that are potentially mobile changes from about 40% (1 to 5 m depth) to 70% (5 m depth). The fraction of potentially mobile munitions in silty clay remains under 20% in both cases.

162

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1



burial depth, m

prob

abili

ty

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1



burial depth, m

prob

abili

ty

Fraction of mobile UXO Fraction of

mobile UXO

Selected Results:Fraction of Mobile Munitions in Silty Clay and Sand

163

As the water depth increases, so does the fraction of potentially mobile munitions. When the water depth is 7 m, about 90% of the munitions in sand and 30% in silty clay are potentially mobile. When the water depth is 10 m, all of the munitions in sand and about 50% of the munitions in silty clay are potentially mobile.

164

0 10 20 300

0.1

0.2

0.3silty claysand

Probability density of shear strength

shear strength, Kpa

Prob

abili

ty d

ensi

ty Silty clay:Lognormal with mean 4.3 kPa and standard deviation 1.8 kPa

Sand:Normal with mean 20 kPa and standard deviation 5 kPa

Shear Strength

165

This slides shows the assumed probability density distributions of shear strength for sand and silty clay used in this analysis.

166

Hydrodynamic PhaseSensitivities

Use computational hydrodynamic model

0 20 40 60 80 10020

40

60

80

100

5 m depth, 50 m/s5 m depth, 300 m/s10 m depth, 50 m/s10 m depth, 300 m/s

impact water angle, degrees

impa

ct se

dim

ent a

ngle

, deg

rees

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

5 m depth, 50 m/s5 m depth, 300 m/s10 m depth, 50 m/s10 m depth, 300 m/s

water impact angle, degrees

ratio

of v

ertic

al v

eloc

ity to

vel

ocity

mag

nitu

de

Estimating Errors in Munition’s Initial Conditions at Mudline

167

This slides gives example results from our sensitivity analysis. We exercised the simplified 2D model of the munition’s propagation in water and sediment to compute the terms in the sensitivity matrix shown in the slide. Selected examples are also given: the munition’s impact angle at the sediment (left) and the magnitude of the vertical component of the munition’s impact velocity at the sediment, normalized by the total magnitude of the munition’s impact velocity at the sediment (right). Both these quantities are plotted as a function of the munition’s impact velocity at the waterline, for various impact conditions.

168

2 4 6 8 10 120

0.05

0.1


prob

abili

ty d

ensi

ty

V00 = 50 m/s, h = 5 m

0 10 20 30 40 50 600

0.05

0.1

0.15


prob

abili

ty d

ensi

ty

V00 = 300 m/s, h = 5 m

4.8 4.9 5 5.10

0.2

0.4


prob

abili

ty d

ensi

ty

V00 = 50 m/s, h = 10m

4 6 8 10 120

0.1

0.2

0.3

0.4


prob

abili

ty d

ensi

ty

V00 = 300 m/s, h = 10 m

ψ isnormallydistributedwithmean45°andstandarddeviation15°

Estimating Velocity Errors in Munition’s Initial Conditions at Mudline

169

Shown on the facing page are selected examples of the distributions of the munition’s impact velocity at the sediment when the munition’s impact angle at the waterline is normally distributed with mean 45 degrees and standard deviation 15 degrees, for several impact conditions.

170

50 60 70 80 900

0.1

0.2

0.3

0.4

0.5


prob

abili

ty d

ensi

ty

V00 = 50 m/s, h = 5 m

50 60 70 800

0.1

0.2

0.3

0.4


prob

abili

ty d

ensi

ty

V00 = 50 m/s, h = 10 m

60 70 80 900

0.1

0.2

0.3


prob

abili

ty d

ensi

ty

V00 = 300 m/s, h = 5 m

30 40 50 60 70 800

0.05

0.1

0.15

0.2


prob

abili

ty d

ensi

ty

V00 = 300 m/s, h = 10 m

Estimating Impact Angle Errors in Munition’s Initial Conditions at Mudline

isnormallydistributedwithmean45°andstandarddeviation15°

171

Shown on the facing page are selected examples of distributions of the munition’s impact angle at the sediment when the munition’s impact angle at the waterline is normally distributed with mean 45 degrees and standard deviation 15 degrees, for several impact conditions.

A-1

References

Aubeny, C., and H. Shi. 2007. “Effect of Rate-Dependent Soil Strength on Cylinders Penetrating Into Soft Clay.” IEEE J. Oceanic Eng. 32 (1) (January).

Chu, P. C., and G. Ray. 2006. “Prediction of High Speed Rigid Body Maneuvering in Air-Water-Sediment Columns. Advances in Fluid Mechanics 6:123–32.

Chu, P. et al. 2001. “Modeling of Underwater Bomb Trajectory for Mine Clearance.” Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 8 (1): 25–36.

Chu, Peter C., and Chenwu Fan. 2007. “Mine-Impact Burial Model (IMPACT35) Verification and Improvement Using Sediment Bearing Factor Method.” IEEE Journal of Oceanic Engineering 32, no. 1 (January).

Holland, K. Todd. 2015. “A Wide Area Risk Assessment Framework for Underwater Military Munitions Response.” MR 2411. Naval Research Laboratory – Stennis Space Center In-Progress Review Meeting, May 20.

Rennie, Sarah, and Alan Brandt. 2015. “Interim Report: Underwater Munitions Expert System to Predict Mobility and Burial, SERDP Project MR-2227.” Laurel, MD: Johns Hopkins University, Applied Physics Laboratory. https://www.serdp-estcp.org/content/download/36174/345934/file/MR-2227-IR.pdf.

Wilson, Jeffrey V., and Alexandra DeVisser. 2009. Final Report: Predicting the Mobility and Burial of Underwater Unexploded Ordnance (UXO) Using the UXO Mobility Model. Contract Report CR-10-012-ENV. ESTCP. November. http://www.dtic.mil/dtic/tr/fulltext/u2/a532814.pdf.

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Jun 2017 – Jul 20174. TITLE AND SUBTITLE


5a. CONTRACT NUMBER HQ0034-14-D-0001

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Teichman, Jeremy A.Macheret, Yevgeny Cazares, Shelley M.

5d. PROJECT NUMBER AM-2-1528

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5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

Institute for Defense Analyses4850 Mark Center Drive Alexandria, VA 22311-1882

8. PERFORMING ORGANIZATION REPORTNUMBER

IDA Document NS D-8616

9. SPONSORING / MONITORING AGENCY NAME(S) ANDADDRESS(ES)

Director and Program Manager for MunitionsSERDP/ESTCP 4800 Mark Center Drive Suite 17D08 Alexandria, VA 22350-3605

10. SPONSOR/MONITOR’S ACRONYM(S)

SERDP/ESTCP

11. SPONSOR/MONITOR’S REPORTNUMBER(S)

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Approved for public release; distribution is unlimited (20 November 2017).

13. SUPPLEMENTARY NOTES

14. ABSTRACTThe Institute for Defense Analyses (IDA) is a not-for-profit company that operates three Federally Funded Research andDevelopment Centers (FFRDCs). We perform scientific and technical analyses for the U.S. government on issues related tonational security. Recently, we performed a numerical analysis for the Strategic Environmental Research and DevelopmentProgram (SERDP) to explore the fidelity of computational models for predicting the initial penetration depth of UnexplodedOrdnance (UXO) in underwater sites. This briefing discusses the details of our analysis. A separate briefing provides ashorter summary.

15. SUBJECT TERMS

accretion, erosion, IMPACT25, IMPACT28, IMPACT35, MINE6-D, scour, sensitivity analysis, STRIKE35, Unexploded Ordnance, UXO 16. SECURITY CLASSIFICATION OF: 17. LIMITATION

OF ABSTRACT

UU

18. NUMBER OF

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171

19a. NAME OF RESPONSIBLE PERSON Nelson, Herbert

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uxo burial prediction fidelity

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